1. Core Vocabulary and Number Relationships

A factor of a positive whole number divides that number exactly, leaving no remainder. A multiple of a number is a product obtained by multiplying it by a whole number. If \(a\times b=n\), then \(a\) and \(b\) are factors of \(n\), and \(n\) is a multiple of both \(a\) and \(b\).

The notation \(a\mid n\) means "\(a\) divides \(n\)." In ordinary fifth grade work, the same idea can be written \(n\div a=k\) with no remainder. For example, \(7\mid63\) because \(63=7\times9\).

Divisibility describes whether one whole number divides another exactly. A factor pair is a pair of whole numbers whose product is the target number. A prime number has exactly two positive factors. A composite number has more than two positive factors.

2. How to Identify Factors

To test whether \(d\) is a factor of \(n\), divide \(n\) by \(d\). If the quotient is a whole number and the remainder is zero, \(d\) is a factor. Multiplication gives an equivalent test: find whether a whole number multiplied by \(d\) equals \(n\).

Every positive whole number has \(1\) and itself as factors. The number \(18\) has factor pairs \(1\times18\), \(2\times9\) and \(3\times6\), so its factors are \(1,2,3,6,9,18\). The list is finite because a positive factor cannot be larger than the positive number it divides, except when fractions or negative numbers are introduced outside this lesson.

A systematic factor search starts at \(1\) and tests possible divisors in order. Record both members of every pair. For \(48\):

• \(48\div1=48\), giving \(1\) and \(48\).
• \(48\div2=24\), giving \(2\) and \(24\).
• \(48\div3=16\), giving \(3\) and \(16\).
• \(48\div4=12\), giving \(4\) and \(12\).
• \(48\div5\) is not whole.
• \(48\div6=8\), giving \(6\) and \(8\).

After \(6\), the next possible divisor is \(7\), which is greater than \(\sqrt{48}\). Any later factor would be paired with a smaller factor already checked. Therefore, the complete factor list is \(1,2,3,4,6,8,12,16,24,48\).

3. Finding Every Factor Pair Efficiently

Factors occur in pairs because multiplication combines two factors to make a product. Listing pairs prevents omissions and duplicates. Begin with \(1\times n\), increase the smaller factor and stop once the factors meet or cross.

For \(72\), the pairs are:

The next test would use \(9\), but \(9\times8\) reverses the pair already listed. Since \(\sqrt{72}\) lies between \(8\) and \(9\), checking possible smaller factors through \(8\) is enough.

Perfect squares have one repeated factor pair. The factor pairs of \(36\) include \(6\times6\). When listing individual factors, write \(6\) once. This gives \(1,2,3,4,6,9,12,18,36\), an odd number of factors. Non-square positive integers have pairs with two different members and therefore an even number of positive factors.

4. How to Identify Multiples

Multiples are generated by multiplication. The positive multiples of \(7\) begin \(7,14,21,28,35,42,\ldots\). If zero is included among whole numbers, \(0=7\times0\) is also a multiple of \(7\). Many classroom lists begin with the first positive multiple, so read the instruction carefully.

A number has infinitely many multiples because the multiplier can keep increasing. Unlike factors, multiples are not limited to values below the starting number. The \(k\)-th positive multiple of \(n\) is \(n\times k\).

To determine whether \(156\) is a multiple of \(12\), divide: \(156\div12=13\). Because the quotient is whole, \(156\) is the thirteenth positive multiple of \(12\). To find the first multiple of \(18\) greater than \(200\), note that \(18\times11=198\) and \(18\times12=216\). The answer is \(216\).

Skip-counting reveals patterns but multiplication notation is more efficient for distant multiples. Instead of writing every multiple of \(24\) until reaching \(600\), calculate \(600\div24=25\). Therefore, \(600\) is the twenty-fifth positive multiple of \(24\).

5. Factors and Multiples Are Two Views of One Equation

Factor and multiple statements reverse perspective. In \(8\times15=120\), \(8\) and \(15\) are factors of \(120\), while \(120\) is a multiple of \(8\) and \(15\). Division confirms both relationships: \(120\div8=15\) and \(120\div15=8\).

The statement "\(4\) is a multiple of \(20\)" is false, but "\(4\) is a factor of \(20\)" is true. The direction of the relationship matters. Use a complete sentence and ask whether the number divides or is produced by multiplication.

Strong multiplication and division skills make these relationships easier. Review fifth grade multiplication and fifth grade division when fact families or exact quotients need reinforcement.

6. Prime, Composite, Zero and One

A prime number has exactly two distinct positive factors: \(1\) and itself. A composite number has more than two positive factors. The number \(13\) is prime because its only positive factors are \(1\) and \(13\). The number \(15\) is composite because \(1,3,5,15\) are factors.

The number \(1\) is neither prime nor composite because it has only one positive factor. The number \(0\) is also neither prime nor composite. Every nonzero whole number divides \(0\), so zero does not have exactly two factors or a finite composite-style factor list.

The number \(2\) is the smallest prime and the only even prime. Every other even number greater than \(2\) is divisible by \(2\), so it has at least \(1,2\) and itself as factors and is composite. Every prime greater than \(2\) is odd, but not every odd number is prime: \(9,15,21\) and \(25\) are odd composite numbers.

To decide whether \(97\) is prime, test possible prime divisors no greater than \(\sqrt{97}\), which is less than \(10\). Test \(2,3,5\) and \(7\). The number is odd, its digit sum \(16\) is not divisible by \(3\), it does not end in \(0\) or \(5\), and \(97\div7\) is not whole. Therefore, \(97\) is prime.

The primes through \(50\) are \(2,3,5,7,11,13,17,19,23,29,31,37,41,43,47\). A sieve model crosses out multiples of each prime to reveal the remaining primes. The dedicated prime numbers lesson extends this classification.

7. Prime Factorization and Factor Trees

Prime factorization writes a composite number as a product of prime factors. Continue splitting composite factors until every factor is prime. Different factor trees produce the same prime factors, apart from order.

For \(84\), one tree begins \(84=7\times12\), then \(12=3\times4\) and \(4=2\times2\). Therefore:

Another tree might begin \(84=6\times14\). Since \(6=2\times3\) and \(14=2\times7\), it reaches the same factorization. The uniqueness of prime factorization means every whole number greater than \(1\) has one prime factorization when order is ignored.

A division ladder uses repeated prime division. For \(180\), divide by \(2\) to get \(90\), by \(2\) to get \(45\), by \(3\) to get \(15\), by \(3\) to get \(5\), and by \(5\) to get \(1\). Thus:

Exponents count repeated factors. The expression \(2^3\times5\) means \(2\times2\times2\times5=40\). Always verify a prime factorization by multiplying. Stop only when every factor is prime; \(84=6\times14\) is a factorization but not a prime factorization.

8. Divisibility Rules for Whole Numbers

Divisibility rules determine whether division will have a remainder without carrying out the full algorithm. They are tests, not estimates. A successful test proves exact divisibility by the specified divisor.

The test for \(6\) requires both \(2\) and \(3\). Being divisible by only one is insufficient. For example, \(25\) has digit sum \(7\) and is odd, so it is divisible by neither \(2\) nor \(3\). The number \(27\) is divisible by \(3\) but not \(2\), so it is not divisible by \(6\).

9. Why Divisibility Rules Work

Understanding the place-value reason behind a rule makes it easier to remember and apply correctly. For divisibility by \(2,5\) and \(10\), all tens, hundreds and larger place values are multiples of \(10\), and therefore multiples of \(2\) and \(5\). Only the ones digit can change the result.

For divisibility by \(4\), every hundred is divisible by \(4\). A whole number can be written as a multiple of \(100\) plus its final two-digit number. Since the multiple of \(100\) is already divisible by \(4\), only the final two digits need testing. For \(7{,}316\):

The rule for \(8\) works similarly because \(1{,}000\) is divisible by \(8\). Everything before the final three digits contributes a multiple of \(1{,}000\), so the final three-digit part determines divisibility.

The digit-sum rules for \(3\) and \(9\) follow from powers of ten leaving remainder \(1\) when divided by \(3\) or \(9\). For example, \(10,100\) and \(1{,}000\) are each one more than a multiple of \(9\). Therefore, \(4{,}572\) has the same remainder upon division by \(9\) as \(4+5+7+2=18\). Because \(18\) is divisible by \(9\), so is \(4{,}572\).

10. Combining Divisibility Tests

Composite divisors can sometimes be tested through smaller coprime factors. A number is divisible by \(6\) exactly when it is divisible by both \(2\) and \(3\). A number is divisible by \(12\) when it is divisible by both \(3\) and \(4\). A number is divisible by \(15\) when it is divisible by both \(3\) and \(5\).

The smaller tests must combine to include every prime factor of the target divisor. Divisibility by \(2\) and \(4\) does not by itself prove divisibility by \(8\), because both tests may be satisfied by the same powers of \(2\). For example, \(12\) is divisible by \(2\) and \(4\), but not by \(8\).

Test \(7{,}560\): it is divisible by \(2\) because it is even, by \(3\) because its digit sum is \(18\), by \(4\) because \(60\) is divisible by \(4\), by \(5\) because it ends in \(0\), by \(6\) because it passes \(2\) and \(3\), by \(8\) because \(560\div8=70\), by \(9\) because the digit sum is \(18\), and by \(10\) because it ends in \(0\).

Combined tests support factor searches. If a number passes tests for \(3\) and \(4\), then \(12\) is a factor. The related quotient can then be found with division. A test confirms the relationship; it does not replace finding the actual group size when a problem asks for it.

11. Common Factors and Greatest Common Factor

A common factor divides each of two or more numbers exactly. The greatest common factor, or GCF, is the largest positive factor shared by all the numbers.

To find the GCF of \(24\) and \(36\), list factors:

• Factors of \(24\): \(1,2,3,4,6,8,12,24\)
• Factors of \(36\): \(1,2,3,4,6,9,12,18,36\)

The common factors are \(1,2,3,4,6,12\), so \(\operatorname{GCF}(24,36)=12\).

GCF problems often involve making the greatest possible number of identical groups with no leftovers. If there are \(24\) red beads and \(36\) blue beads, the greatest number of identical packs is \(12\). Each pack contains \(24\div12=2\) red beads and \(36\div12=3\) blue beads.

The word "greatest" refers to the common factor, not the size of each resulting object automatically. Label the context. Sometimes the GCF gives the number of groups; in rectangle tiling, it may give the largest square side length.

12. Common Multiples and Least Common Multiple

A common multiple is a multiple shared by two or more numbers. The least common multiple, or LCM, is the smallest positive common multiple.

For \(8\) and \(12\):

• Positive multiples of \(8\): \(8,16,24,32,40,48,\ldots\)
• Positive multiples of \(12\): \(12,24,36,48,\ldots\)

The first positive common multiple is \(24\), so \(\operatorname{LCM}(8,12)=24\). Other common multiples, such as \(48,72\) and \(96\), are multiples of the LCM.

LCM problems often involve repeating schedules, matching cycle lengths or finding a common denominator. If one light flashes every \(8\) seconds and another every \(12\) seconds, they flash together every \(24\) seconds after starting together.

If one number is a factor of the other, the larger number is their LCM. Since \(6\mid30\), \(\operatorname{LCM}(6,30)=30\). Their GCF is the smaller number, \(6\).

13. Using Prime Factors for GCF and LCM

Prime factorization gives a structured method for larger numbers. Write each number as prime powers. For the GCF, select only primes shared by every number and use the smallest exponent. For the LCM, include every prime that appears and use the greatest exponent.

For \(72\) and \(90\):

The shared primes are \(2\) and \(3\). Use \(2^1\) and \(3^2\) for the GCF:

For the LCM, use \(2^3,3^2\) and \(5\):

A useful check for two positive whole numbers is:

Here \(18\times360=6{,}480\) and \(72\times90=6{,}480\), so the results agree.

14. Connections to Fractions

Common factors simplify fractions. In \(\frac{18}{30}\), the GCF of \(18\) and \(30\) is \(6\). Divide numerator and denominator by \(6\):

Dividing both parts by the same nonzero factor preserves the fraction's value. If numerator and denominator have no common factor greater than \(1\), the fraction is in simplest form.

Common multiples help add or compare fractions with different denominators. For \(\frac38\) and \(\frac5{12}\), the LCM of \(8\) and \(12\) is \(24\), so \(24\) is the least common denominator. Convert \(\frac38=\frac9{24}\) and \(\frac5{12}=\frac{10}{24}\). Therefore, \(\frac5{12}>\frac38\).

Factor reasoning also supports fraction multiplication and division. Identifying common factors before multiplying can simplify calculations, while multiples help understand equivalent fractions. Continue with fractions and mixed numbers, adding and subtracting fractions and understanding fraction multiplication.

15. Patterns in Factors and Multiples

Patterns can predict factor structure. Every even number has \(2\) as a factor. Every positive multiple of \(10\) is also a multiple of \(2\) and \(5\). Every multiple of \(6\) is a multiple of both \(2\) and \(3\). However, reversing these statements requires care: a multiple of \(2\) is not necessarily a multiple of \(6\).

Consecutive whole numbers have GCF \(1\). Any common factor of \(n\) and \(n+1\) would also divide their difference, which is \(1\). Therefore, no common factor greater than \(1\) is possible. For example, the factors of \(20\) and \(21\) overlap only at \(1\).

Two different prime numbers have GCF \(1\) and LCM equal to their product. For primes \(7\) and \(11\), \(\operatorname{GCF}=1\) and \(\operatorname{LCM}=77\). A prime and one of its multiples have GCF equal to the prime.

Perfect squares contain paired prime factors. In prime factorization, every exponent of a perfect square is even. For example, \(144=2^4\times3^2\). This explains why its factor pairs include \(12\times12\) and why it has an odd number of factors.

Explore related sequences in the fifth grade number patterns lesson.

16. Mathematical Reasoning and Short Proofs

Factor reasoning is stronger when a claim is justified rather than checked through a few examples. Consider: "The sum of two multiples of \(5\) is a multiple of \(5\)." If the numbers are \(5a\) and \(5b\), their sum is \(5a+5b=5(a+b)\). Because \(a+b\) is a whole number, the sum is a multiple of \(5\).

The product of a multiple of \(4\) and any whole number is also a multiple of \(4\). If the first number is \(4a\), multiplying by \(b\) gives \(4ab=4(ab)\). This proof applies to every whole-number choice, not only selected examples.

The sum of two odd numbers is even. Write the odd numbers as \(2a+1\) and \(2b+1\). Their sum is \(2a+2b+2=2(a+b+1)\), which has factor \(2\). Similarly, the product of two odd numbers is odd because \((2a+1)(2b+1)\) simplifies to \(2(2ab+a+b)+1\).

A common misconception says every larger number has more factors. Compare \(24\) and \(29\). The number \(24\) has eight positive factors, while the larger number \(29\) is prime and has only two. Size alone does not determine factor count.

Counterexamples disprove universal claims. The statement "All odd numbers are prime" is disproved by \(9=3\times3\). The statement "Every multiple of \(4\) is a multiple of \(8\)" is disproved by \(12\). One valid counterexample is enough to show an always-claim is false.

Count factors from prime exponents

Prime factorization can determine how many positive factors a number has without listing every pair. Consider \(72=2^3\times3^2\). A factor of \(72\) may use zero, one, two or three factors of \(2\), giving \(4\) choices. It may use zero, one or two factors of \(3\), giving \(3\) choices. Each choice for the power of \(2\) can pair with each choice for the power of \(3\), so \(72\) has \(4\times3=12\) positive factors.

The same idea extends when there are more prime factors: add \(1\) to each exponent and multiply the results. For \(180=2^2\times3^2\times5^1\), the number of positive factors is \((2+1)(2+1)(1+1)=3\times3\times2=18\).

This method also explains perfect-square patterns. A positive integer has an odd number of positive factors exactly when it is a perfect square. Factors usually pair as two different numbers. In a square, one pair repeats at the square root, so it contributes one unpaired factor. In prime factorization, a perfect square has only even exponents. Adding \(1\) to even exponents produces odd factor counts, and a product of odd counts remains odd.

For example, \(144=2^4\times3^2\), so it has \((4+1)(2+1)=15\) positive factors. The middle pair is \(12\times12\). By contrast, \(72=2^3\times3^2\) is not a square because one exponent is odd, and its \(12\) positive factors form six distinct pairs.

Solve missing-digit divisibility puzzles

Divisibility rules can determine unknown digits without testing every complete number. In \(4\Box2\), the number is divisible by \(3\) when \(4+\Box+2=6+\Box\) is a multiple of \(3\). The possible digits are \(0,3,6,9\), producing \(402,432,462,492\).

If \(4\Box2\) must be divisible by \(6\), it must pass the tests for both \(2\) and \(3\). The final digit \(2\) already makes every version even, so the same four digits \(0,3,6,9\) work. If the number must be divisible by \(9\), then \(6+\Box\) must be \(9\) or \(18\). The possible digits are \(3\) and \(12\), but \(12\) is not a single digit, so only \(3\) works.

For a test involving \(4\), focus on the final two digits. In \(73\Box\), the final two-digit number is \(30+\Box\). Among \(30,31,\ldots,39\), the multiples of \(4\) are \(32\) and \(36\), so the missing digit can be \(2\) or \(6\). Earlier digits do not affect this test because every hundred is divisible by \(4\).

Several conditions can be combined. Find digits making \(5\Box0\) divisible by both \(6\) and \(9\). The number is already even, so divisibility by \(6\) requires digit sum \(5+\Box\) to be divisible by \(3\). Divisibility by \(9\) requires the same sum to be a multiple of \(9\). With a single digit, \(5+\Box=9\), so \(\Box=4\). The number \(540\) passes both conditions.

Check the completed number after finding a digit. A digit-sum condition may solve divisibility by \(3\) or \(9\), but another required condition may involve the final digits. Written evidence should name every test used.

Decide which relationship a problem requires

Factors, multiples, GCF and LCM solve different questions. Before calculating, identify whether the problem asks for possible equal arrangements, the greatest identical grouping, repeated events meeting, or a common unit.

Language alone is not enough. "Largest" can signal GCF when asking for largest tile size, but "smallest number of items that can be arranged in either group size" signals LCM. Translate the situation into a requirement: must the answer divide the given numbers, or must the given numbers divide the answer?

For example, a \(48\)-centimeter ribbon and a \(60\)-centimeter ribbon are cut into equal pieces of the greatest possible length. The unknown length must divide both \(48\) and \(60\), so use the GCF, which is \(12\). If machines repeat every \(48\) and \(60\) minutes, the unknown meeting time must be a multiple of both, so use the LCM, which is \(240\) minutes.

17. Practical Applications and Word Problems

Arrange objects in equal rows

A teacher has \(48\) chairs and wants equal rows with no chairs left. Every factor of \(48\) is a possible row count: \(1,2,3,4,6,8,12,16,24,48\). A requirement of at least \(4\) rows and at least \(4\) chairs per row narrows the possibilities to \(4\times12\), \(6\times8\), \(8\times6\) and \(12\times4\).

Make the greatest number of identical groups

A club has \(42\) notebooks and \(56\) pens. To make the greatest number of identical kits with no leftovers, find \(\operatorname{GCF}(42,56)=14\). Each kit has \(42\div14=3\) notebooks and \(56\div14=4\) pens.

Find when schedules coincide

One bus arrives every \(18\) minutes and another every \(24\) minutes. Starting together, they next arrive together after \(\operatorname{LCM}(18,24)=72\) minutes.

Tile a rectangle with the largest equal squares

A rectangle measures \(84\) centimeters by \(60\) centimeters. The largest square tile side that fits both dimensions exactly is \(\operatorname{GCF}(84,60)=12\) centimeters. The layout uses \(84\div12=7\) tiles by \(60\div12=5\) tiles, or \(35\) tiles.

Determine whether equal sharing is possible

A school wants to distribute \(1{,}134\) tickets equally among \(9\) classes. The digit sum is \(1+1+3+4=9\), so the total is divisible by \(9\). Each class receives \(1{,}134\div9=126\) tickets.

Solve multi-step situations

A store has \(72\) red folders and \(90\) blue folders. It creates the greatest possible number of identical packs, then places \(3\) packs in each carton. The GCF is \(18\), so there are \(18\) packs. Each pack contains \(4\) red and \(5\) blue folders. Dividing \(18\) packs by \(3\) gives \(6\) cartons. See the multi-step word problems lesson for more mixed-operation applications.

18. Interactive Factor and Prime Analyzer

Enter a positive whole number. The analyzer lists its factors and factor pairs, classifies it as prime or composite, writes its prime factorization and reports which standard divisibility tests it passes.

19. Interactive Common-Factor and Common-Multiple Explorer

Enter two positive whole numbers to compare their factor and multiple structure. The tool reports common factors, GCF and LCM and verifies the two-number product identity.

20. Interactive Practice Generator

Generate a factor, multiple, divisibility or classification question. Answer before using the hint or reveal controls.

21. Fully Worked Examples

Worked example 1: List factors

Question: Find every factor of \(96\).

Solution: Factor pairs are \(1\times96\), \(2\times48\), \(3\times32\), \(4\times24\), \(6\times16\) and \(8\times12\). Stop before reversing pairs.

Answer: \(1,2,3,4,6,8,12,16,24,32,48,96\).

Worked example 2: Decide whether a number is prime

Question: Is \(91\) prime?

Solution: \(\sqrt{91}<10\), so test primes \(2,3,5,7\). The number is odd, digit sum \(10\) is not divisible by \(3\), and it does not end in \(0\) or \(5\). But \(91\div7=13\).

Answer: Composite, because \(91=7\times13\).

Worked example 3: Prime factorization

Question: Write the prime factorization of \(360\).

Solution: \(360=36\times10=(2^2\times3^2)(2\times5)\).

Answer: \(360=2^3\times3^2\times5\).

Worked example 4: Apply several divisibility rules

Question: Which of \(2,3,4,5,6,8,9,10\) divide \(6{,}480\)?

Solution: It is even; digit sum is \(18\); final two digits \(80\) divide by \(4\); it ends in \(0\); final three digits \(480\) divide by \(8\). Therefore, every listed divisor divides \(6{,}480\).

Worked example 5: Find GCF

Question: Find \(\operatorname{GCF}(54,72)\).

Solution: \(54=2\times3^3\) and \(72=2^3\times3^2\). Use shared primes with smaller exponents: \(2^1\times3^2=18\).

Answer: \(18\).

Worked example 6: Find LCM

Question: Find \(\operatorname{LCM}(18,24)\).

Solution: \(18=2\times3^2\) and \(24=2^3\times3\). Use greatest exponents: \(2^3\times3^2=72\).

Answer: \(72\).

Worked example 7: Simplify a fraction

Question: Simplify \(\frac{42}{56}\).

Solution: \(\operatorname{GCF}(42,56)=14\). Divide numerator and denominator by \(14\).

Answer: \(\frac{42}{56}=\frac34\).

Worked example 8: Recurring schedules

Question: Two alarms sound every \(15\) and \(20\) minutes. If they sound together at 9:00, when do they next sound together?

Solution: \(\operatorname{LCM}(15,20)=60\).

Answer: \(60\) minutes later, at 10:00.

22. Common Mistakes and How to Fix Them

23. Independent Practice

Show factor pairs, divisibility evidence or prime factorizations. A yes-or-no answer should include a short mathematical reason.

Part A: Factors and multiples

1. List every factor of \(40\).
2. List every factor pair of \(63\).
3. Write the first six positive multiples of \(14\).
4. Find the first multiple of \(18\) greater than \(250\).
5. Is \(17\) a factor of \(221\)? Explain.
6. Is \(315\) a multiple of \(14\)? Explain.

Part B: Prime, composite and factorization

7. Classify \(1\), \(2\), \(39\) and \(53\).
8. Explain why \(87\) is composite.
9. Write the prime factorization of \(72\).
10. Write the prime factorization of \(150\).
11. Find a composite number between \(40\) and \(50\) with exactly three positive factors.
12. Explain why every even number greater than \(2\) is composite.

Part C: Divisibility

13. Which of \(2,3,4,5,6,8,9,10\) divide \(3{,}240\)?
14. Is \(7{,}434\) divisible by \(3\)? By \(9\)?
15. Find the smallest digit that makes \(52\Box\) divisible by \(3\).
16. Find all digits that make \(7\Box5\) divisible by \(9\).
17. Is \(18{,}632\) divisible by \(8\)? Explain.
18. Explain why a number ending in \(5\) cannot be divisible by \(2\).

Part D: GCF and LCM

19. Find \(\operatorname{GCF}(36,48)\).
20. Find \(\operatorname{LCM}(12,30)\).
21. Find the GCF and LCM of \(45\) and \(60\).
22. Find \(\operatorname{GCF}(84,126)\).
23. Find \(\operatorname{LCM}(16,24)\).
24. Verify the GCF-LCM product identity for \(18\) and \(30\).

Part E: Applications

25. A teacher has \(32\) pencils and \(48\) erasers. What is the greatest number of identical kits that can be made with no leftovers?
26. Two bells ring every \(14\) and \(20\) minutes. If they ring together now, after how many minutes will they next ring together?
27. A \(72\)-by-\(108\) centimeter rectangle is tiled with the largest possible equal square tiles. Find the tile side length and number of tiles.
28. Simplify \(\frac{45}{75}\) using the GCF.
29. Find the least common denominator of \(\frac5{12}\) and \(\frac7{18}\).
30. A shop has \(60\) juice boxes, \(84\) granola bars and \(108\) pieces of fruit. Find the greatest number of identical snack packs and the contents of each.

24. Answers and Explanations

25. Study, Teaching and Review Guidance

For students: build relationships, not separate lists

Connect every factor fact to multiplication and division. When you write \(6\times14=84\), also state that \(6\) and \(14\) are factors, \(84\) is their multiple, and \(84\) is divisible by each. This network of statements reduces memorization.

Mix tasks in short sessions: one complete factor list, one prime classification, one factor tree, two divisibility tests, one GCF, one LCM and one application. Check factor lists through pair products and check GCF-LCM answers through the product identity when there are two numbers.

Keep an error log with categories such as missing factor pair, repeated factor, factor-versus-multiple reversal, incorrect prime classification, unfinished factorization, incomplete divisibility test and GCF-versus-LCM choice.

For parents and tutors: ask structural questions

Ask, "Which multiplication fact proves that?", "Have the factor pairs crossed?", "Why is the remainder zero?", "Does this problem ask for the greatest grouping size or the first shared time?" and "Can you disprove the claim with a counterexample?" These questions reveal understanding without supplying results.

For teachers: move among arrays, lists and symbolic forms

Use rectangular arrays to show factor pairs, number lines for multiples and Venn diagrams for common factors. Connect each model to equations. Require students to explain why factor searches stop at the square root and why GCF and LCM solve different contexts.

Use fifth grade math worksheets for printable review. Continue with mixed whole-number operations, numerical expressions and fractions and mixed numbers.

26. Frequently Asked Questions