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Divisibility &
Factors

Master the rules of divisibility and learn to find all factors of any integer: essential skills for the ISEE Quantitative Reasoning section.

ISEE Upper Level

What Does Divisibility Mean?

An integer \(a\) is divisible by an integer \(b\) (where \(b \neq 0\)) if there exists an integer \(k\) such that:

For example, \(24\) is divisible by \(6\) because \(24 = 6 \times 4\) with no remainder.

Factors vs. Multiples

Factor
A number that divides evenly into another. Factors of \(12\): \(1, 2, 3, 4, 6, 12\).
Multiple
A number you get by multiplying by an integer. Multiples of \(5\): \(5, 10, 15, 20, \ldots\)

Factor Pairs of \(36\)

36 1 36 2 18 3 12 4 9 6 6 Factor Partner

The Divisibility Rules

Divisible by \(2\)
Last digit is even (\(0,2,4,6,8\)).
Divisible by \(3\)
Sum of digits is divisible by \(3\).
Divisible by \(4\)
Last two digits form a number divisible by \(4\).
Divisible by \(5\)
Last digit is \(0\) or \(5\).

More Divisibility Rules

Divisible by \(6\)
Divisible by both \(2\) and \(3\).
Divisible by \(8\)
Last three digits form a number divisible by \(8\).
Divisible by \(9\)
Sum of digits is divisible by \(9\).
Divisible by \(10\)
Last digit is \(0\).

Prime Factorization: The Master Key

Every integer greater than \(1\) can be written as a unique product of prime factors. This is called the Fundamental Theorem of Arithmetic.

360 36 10 6 6 2 5 2 3 2 3
\(360 = 2^3 \times 3^2 \times 5\)

Finding All Factors Systematically

If the prime factorization of \(n\) is \(p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_k^{a_k}\), then the total number of factors is:

Factors vs. Prime Factors

Factors Prime Factors
Definition All integers that divide evenly Only the prime numbers that divide evenly
Of \(12\) \(1,2,3,4,6,12\) \(2, 3\)
Of \(45\) \(1,3,5,9,15,45\) \(3, 5\)
Count Finite, can be large Usually small (2-4 primes)
Purpose Simplifying, GCF, LCM Fundamental decomposition

GCF and LCM from Factors

GCF
Greatest Common Factor: Take the lowest power of each shared prime. GCF\((12, 18) = 2^1 \times 3^1 = 6\).
LCM
Least Common Multiple: Take the highest power of each prime. LCM\((12, 18) = 2^2 \times 3^2 = 36\).

How to Apply Divisibility & Factor Rules

1
Read carefully: Does the problem ask for divisibility, factors, prime factors, GCF, or LCM?
2
Quick divisibility check: Use the rules for \(2, 3, 4, 5, 6, 8, 9, 10\) to eliminate or confirm divisors.
3
Prime factorize: Break the number into prime factors using a factor tree.
4
Count or combine: Use the factor-count formula, or compute GCF/LCM from the prime factorizations.
5
Verify: Multiply back to confirm your answer.

Example 1: Divisibility Check

Problem: Which of the following numbers is divisible by both \(3\) and \(4\)?
\(126, \; 148, \; 156, \; 162\)

1
Check divisibility by \(3\) (digit sum divisible by \(3\)):
\(126 \to 1+2+6=9\) ✓ | \(148 \to 1+4+8=13\) ✕ | \(156 \to 1+5+6=12\) ✓ | \(162 \to 1+6+2=9\) ✓
2
Check divisibility by \(4\) (last two digits divisible by \(4\)):
\(126 \to 26 \div 4 = 6.5\) ✕ | \(156 \to 56 \div 4 = 14\) ✓ | \(162 \to 62 \div 4 = 15.5\) ✕
3
Only \(156\) passes both tests. Verify: \(156 = 12 \times 13\), and \(12 = 3 \times 4\). ✓

Example 2: Counting Factors

Problem: How many positive factors does \(180\) have?

1
Prime factorize: \(180 = 2^2 \times 3^2 \times 5^1\)
2
Apply the formula: \((2+1)(2+1)(1+1) = 3 \times 3 \times 2\)
3
Answer: \(18\) positive factors.

Example 3: GCF and LCM

Problem: Find the GCF and LCM of \(48\) and \(72\).

1
Prime factorize both:
\(48 = 2^4 \times 3^1\)
\(72 = 2^3 \times 3^2\)
2
GCF (take the lower power of each shared prime):
\(\text{GCF} = 2^3 \times 3^1 =\) \(24\)
3
LCM (take the higher power of each prime):
\(\text{LCM} = 2^4 \times 3^2 =\) \(144\)
4
Verify: \(24 \times 144 = 3456 = 48 \times 72\). ✓

Common Mistakes to Avoid

✕ Forgetting \(1\) and the Number Itself
Every positive integer has \(1\) and itself as factors. When listing factors, always include both endpoints.
✕ Confusing GCF with LCM
GCF uses the lower power, LCM uses the higher power. Mixing them up gives the wrong answer every time.
✕ Stopping the Factor Tree Too Early
Don't stop at composite numbers like \(4\) or \(9\). Keep breaking down until every leaf is prime.

Quick Reference Summary

Divisor Rule Example (\(7{,}920\))
\(2\) Last digit even \(0\) is even ✓
\(3\) Digit sum divisible by \(3\) \(7+9+2+0=18\) ✓
\(4\) Last two digits ÷ \(4\) \(20 \div 4 = 5\) ✓
\(5\) Last digit \(0\) or \(5\) Ends in \(0\) ✓
\(6\) Both \(2\) and \(3\) Both pass ✓
\(8\) Last three digits ÷ \(8\) \(920 \div 8 = 115\) ✓
\(9\) Digit sum divisible by \(9\) \(18 \div 9 = 2\) ✓
\(10\) Last digit \(0\) Ends in \(0\) ✓

Question 1 of 5

How many positive factors does \(72\) have?

Question 2 of 5

Which of the following is not divisible by \(6\)?

Question 3 of 5

What is the GCF of \(84\) and \(126\)?

Question 4 of 5

If \(n = 2^3 \times 5^2 \times 7\), which of the following is not a factor of \(n\)?

Question 5 of 5

The LCM of two numbers is \(120\) and their GCF is \(12\). If one number is \(24\), what is the other number?