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Divisibility &
Factors

Master the art of breaking numbers apart: learn divisibility rules, find all factors, and reason about how numbers relate on the ISEE.

ISEE Middle Level

What Is Divisibility?

We say that an integer \( a \) is divisible by another integer \( b \) when dividing leaves no remainder. In symbols:

If there is a remainder, the number is not divisible. For instance, \( 25 \div 6 = 4 \) R \( 1 \), so 25 is not divisible by 6.

What Are Factors?

A factor of a number \( n \) is any whole number that divides \( n \) evenly. Factors always come in pairs that multiply to give \( n \).

24 1 24 2 12 3 8
\( 1 \times 24 \)
(also \( 4 \times 6 \))
\( 3 \times 8 \)

Why It Matters on the ISEE

Quantitative Comparisons
Many QC problems ask you to compare factor counts or decide which column is greater based on divisibility.
Word Problems
Questions about grouping items evenly, splitting objects into equal sets, or "How many ways?" depend on factor reasoning.
Speed Advantage
Knowing divisibility rules lets you eliminate wrong answers instantly without long division.

Key Divisibility Rules

2
Divisible by 2: the last digit is even (0, 2, 4, 6, or 8).
3
Divisible by 3: the sum of digits is divisible by 3.
4
Divisible by 4: the last two digits form a number divisible by 4.
5
Divisible by 5: the last digit is 0 or 5.
6
Divisible by 6: divisible by both 2 and 3.
9
Divisible by 9: the sum of digits is divisible by 9.
10
Divisible by 10: the last digit is 0.

Testing \( 342 \) for Divisibility

342 Div by 2? Last digit: 2 (even) ✓ YES Div by 3?
\( 3+4+2 = 9 \) (div by 3)
✓ YES Div by 5? Last digit: 2 (not 0/5) ✕ NO Div by 6? Both 2 and 3 ✓ → YES Div by 9?
\( 3+4+2 = 9 \) → YES

Factors vs. Multiples

Factors Multiples
Definition Numbers that divide into \( n \) Numbers you get by multiplying \( n \)
Count Finite list Infinite list
Size \( \leq n \) \( \geq n \)
Example for 12 1, 2, 3, 4, 6, 12 12, 24, 36, 48, ...
Key question "What divides 12?" "What does 12 divide?"

How to Find All Factors

1
Start at 1. Write the pair: \( 1 \times n = n \). So 1 and \( n \) are always factors.
2
Test 2, 3, 4, ... in order. For each, check: does it divide \( n \) evenly? If yes, record both the divisor and the quotient.
3
Stop when pairs repeat. Once the divisor meets or exceeds the quotient, you have found every factor.
4
List in order. Arrange from smallest to largest for a clean answer.

Example: Factors of \( 36 \)

Divisor Quotient Pair 1 36 (1, 36) 2 18 (2, 18) 3 12 (3, 12) 4 9 (4, 9) 6 6 (6, 6) STOP: divisor = quotient

Example: Is \( 2{,}748 \) Divisible by 6?

1
Check divisibility by 2: the last digit is 8, which is even. ✓
2
Check divisibility by 3: sum of digits: \( 2 + 7 + 4 + 8 = 21 \). Is 21 divisible by 3? \( 21 \div 3 = 7 \). ✓
3
Conclusion: since 2,748 is divisible by both 2 and 3, it is divisible by 6.

Example: How Many Factors Does \( 60 \) Have?

First, find the prime factorization: \( 60 = 2^2 \times 3 \times 5 \).

For \( 60 = 2^2 \times 3^1 \times 5^1 \):

The 12 factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.

GCF and LCM Connection

GCF
Greatest Common Factor: the largest number that is a factor of both numbers. Find by listing shared factors or using prime factorization (take the lowest powers).
LCM
Least Common Multiple: the smallest number that is a multiple of both numbers. Use prime factorization (take the highest powers).

Example: GCF and LCM of \( 18 \) and \( 24 \)

18 24 3 2 3 2 2
\( 18 = 2 \times 3 \times 3 \)
\( 24 = 2 \times 2 \times 2 \times 3 \)
GCF
Shared primes: \( 2 \times 3 = \) \(6\)
LCM
All primes (highest powers): \( 2^3 \times 3^2 = \) \(72\)

Common Mistakes

✕ Forgetting 1 and \( n \)
Every number has 1 and itself as factors. Students often skip these when listing factors, leading to an incomplete count.
✕ Confusing Factors and Multiples
Factors are smaller than or equal to \( n \). Multiples are greater than or equal to \( n \). Mixing them up is the most common ISEE trap.
✕ Missing Factor Pairs
When testing divisors, always record both the divisor and the quotient. For example, if \( 3 \mid 36 \), you get factors 3 and 12.

Summary of Key Rules

Divisor Rule Quick Example
2 Last digit even 48 → 8 is even ✓
3 Digit sum div by 3 141 → \(1+4+1=6\) ✓
4 Last 2 digits div by 4 316 → \(16 \div 4=4\) ✓
5 Last digit 0 or 5 735 → ends in 5 ✓
6 Div by both 2 and 3 42 → even + \(4+2=6\) ✓
9 Digit sum div by 9 729 → \(7+2+9=18\) ✓
10 Last digit 0 250 → ends in 0 ✓

Question 1 of 5

Which of the following numbers is divisible by both 3 and 4?

Question 2 of 5

How many factors does \( 48 \) have?

Question 3 of 5

What is the greatest common factor (GCF) of \( 36 \) and \( 54 \)?

Question 4 of 5

A teacher has 72 pencils and wants to divide them equally among bags with no pencils left over. Which of the following is NOT a possible number of bags?

Question 5 of 5

If \( n \) is a positive integer and \( n \) is divisible by both 6 and 10, what is the smallest possible value of \( n \)?