Factors

A factor of a number divides into it exactly with no remainder. Any whole number divided by 1 gives itself — for instance, \(5 \div 1 = 5\), \(100 \div 1 = 100\), \(1{,}000{,}000 \div 1 = 1{,}000{,}000\). Since this always produces a whole number, 1 is always a factor.

Students often skip 1 when listing factors because it feels too obvious. But 1 × the number itself is always a valid factor pair, so 1 must always be included.

The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36 — that’s 9 factors. The factors of 64 are: 1, 2, 4, 8, 16, 32, 64 — that’s 7 factors. So 36 has two more factors than 64, even though 64 is the larger number.

This challenges the misconception that bigger numbers always have more factors. The number of factors depends on how a number is built from primes, not on how large it is. 36 = 2² × 3² has factors from two different primes, while 64 = 2⁶ is a power of a single prime.

An even number is defined as a number that can be divided exactly by 2. That means \(n \div 2\) gives a whole number — which is exactly the test for 2 being a factor. For example, \(14 \div 2 = 7\), \(100 \div 2 = 50\), \(2 \div 2 = 1\). Since 2 always divides in exactly, 2 is always a factor.

This connection works both ways: “even” and “has 2 as a factor” mean precisely the same thing. Students who understand this can use it as a shortcut — for instance, they can immediately say that 2 is a factor of 378 without doing the long division, simply because 378 is even.

We need to check every whole number from 1 to 7. \(7 \div 1 = 7\) ✓, \(7 \div 2 = 3.5\) ✗, \(7 \div 3 = 2.33…\) ✗, \(7 \div 4 = 1.75\) ✗, \(7 \div 5 = 1.4\) ✗, \(7 \div 6 = 1.16…\) ✗, \(7 \div 7 = 1\) ✓. Only 1 and 7 divide exactly, so 7 has exactly two factors.

Some students say prime numbers have “no factors” or “one factor”. In fact, every prime has exactly two factors: 1 and itself. That’s the definition of a prime number — a whole number greater than 1 with exactly two factors.

Example: 4 (factors: 1, 2, 4)

Another: 9 (factors: 1, 3, 9)

Creative: 49 (factors: 1, 7, 49) — or 169 (= 13²). The only numbers with exactly three factors are squares of prime numbers.

Trap: 6 — a student might list 1, 2, 3 and think that’s three factors, but they’ve forgotten to include 6 itself. The factors of 6 are 1, 2, 3, and 6 — that’s four factors.

Example: 11 (factors: 1 and 11)

Another: 17 (factors: 1 and 17)

Creative: 97 — the largest two-digit prime. Or 41, 59, 67 — any prime greater than 10 works.

Trap: 15 — students often assume odd numbers are prime. But \(15 = 3 \times 5\), so its factors are 1, 3, 5, and 15. Being odd doesn’t mean a number is prime.

Example: 18 (factors: 1, 2, 3, 6, 9, 18)

Another: 28 (factors: 1, 2, 4, 7, 14, 28)

Creative: 45 (factors: 1, 3, 5, 9, 15, 45) — students tend to pick even numbers, so an odd one catches people off guard. Others include 12, 20, 32, and 44.

Trap: 24 — a student might list 1, 2, 3, 4, 6, 24 and count six, but they’ve missed 8 and 12. The full list is 1, 2, 3, 4, 6, 8, 12, 24 — that’s eight factors. Not checking all factor pairs is a common error.

Example: 24 (since \(120 \div 24 = 5\))

Another: 60 (since \(120 \div 60 = 2\))

Creative: 120 itself — students often forget that a number is always a factor of itself. Or 40 (since \(120 \div 40 = 3\)). The factors of 120 greater than 20 are: 24, 30, 40, 60, and 120.

Trap: 240 — a student might see “greater than 20” and “120” and pick a multiple instead of a factor. Another trap is 25 — since both 120 and 25 end in 0 or 5, a student might assume 25 divides into 120. But \(120 \div 25 = 4.8\), which isn’t a whole number.

Most numbers have factors that come in pairs: for 12, the pairs are 1 × 12, 2 × 6, 3 × 4, giving 6 factors (even). But square numbers have an odd number of factors because one factor pair uses the same number twice: for 9, the pairs are 1 × 9 and 3 × 3, giving just 3 factors (odd).

True case: 10 has 4 factors (1, 2, 5, 10). False case: 16 has 5 factors (1, 2, 4, 8, 16).

The factors of 12 are 1, 2, 3, 4, 6, and 12. Since \(60 = 5 \times 12\), any number that divides exactly into 12 must also divide exactly into 60. Check: \(60 \div 1 = 60\) ✓, \(60 \div 2 = 30\) ✓, \(60 \div 3 = 20\) ✓, \(60 \div 4 = 15\) ✓, \(60 \div 6 = 10\) ✓, \(60 \div 12 = 5\) ✓.

The key principle: if a is a factor of b, and b is a factor of c, then a must also be a factor of c. Since 12 is itself a factor of 60, all of 12’s factors “carry through” to 60.

If a number is odd, none of its factors can be even. Suppose an even number did divide into an odd number — then we could write: odd number = even factor × something. But an even number multiplied by any whole number always gives an even result, which contradicts the number being odd.

For example, the factors of 15 are 1, 3, 5, 15 — all odd. The factors of 27 are 1, 3, 9, 27 — all odd. An odd number can only ever have odd factors.

Most factors of a number are less than it — for example, the factors of 20 that are less than 20 are 1, 2, 4, 5, and 10. But every number is a factor of itself, since \(20 \div 20 = 1\) exactly. So 20 itself is a factor of 20, and it is certainly not “less than” 20.

True case: 6 is a factor of 18 and 6 < 18. False case: 18 is a factor of 18 and 18 is not less than 18. This is a common source of error when students list factors — they stop before reaching the number itself, because they assume factors must be smaller.

You cannot divide a number by 0. A factor must divide exactly into a number to produce a whole number result, but dividing by zero is undefined. Therefore, 0 is never a factor of any non-zero number.

This flips our understanding of division on its head! A number \(n\) is a factor of 0 if \(0 \div n\) gives a whole number. Since \(0 \div n = 0\) for any number greater than 0, every whole number is indeed a factor of 0.

The student has forgotten 1 and 18 — two of the most commonly missed factors. Students often skip these because they feel “too obvious” or because they don’t think of a number being a factor of itself.

The complete list of factors of 18 is: 1, 2, 3, 6, 9, 18. Working in factor pairs avoids this mistake: 1 × 18, 2 × 9, 3 × 6. Every factor pair automatically includes both 1 and the number itself.

The answer is correct — 5 is a factor of 30 — but the reasoning is dangerously wrong. The student believes “smaller means factor”, which is false. A counterexample breaks this immediately: 7 is smaller than 30 but \(30 \div 7 = 4.28…\), so 7 is not a factor of 30.

The correct test is divisibility: does \(30 \div 5\) give a whole number? Yes — \(30 \div 5 = 6\) — so 5 is a factor. Also, a number is always a factor of itself (\(30 \div 30 = 1\)), so factors aren’t always “smaller” in the way the student means.

The student has confused factors with multiples. The numbers 10, 20, 30, 40, 50 are the first five multiples of 10. Multiples go up — they are found by multiplying. Factors go into — they are found by dividing.

The factors of 10 are the numbers that divide exactly into 10: 1, 2, 5, 10. That’s only four factors, not five — so the question itself is a clue that something has gone wrong. Factors are a finite set; multiples go on forever.

The student is double-counting 7 because it appears twice in the factor pair \(7 \times 7\). But a factor is simply a number that divides in exactly — we list each value once regardless of how it appears in factor pairs. The number 7 divides into 49 giving 7, and that’s one factor, listed once.

The factors of 49 are 1, 7, 49 — just three factors. This is a property of all square numbers: the repeated factor pair means they have an odd number of factors.

The student has listed all the factors, not just the prime factors. This is a very common exam error.

To be a prime factor, the number must be a factor of 20 AND be a prime number itself. 1 is not prime. 4, 10, and 20 are composite. The only prime factors of 20 are 2 and 5.