Why it matters on the HSPT
Questions about factors, multiples, and primes look like trivia but decide a steady handful of points. They appear straight, as in which of these is prime, and they hide inside word problems: the largest number of equal groups, the first time two events happen together, the common denominator a fraction sum needs. Divisibility rules also speed up everything else on the test, because they tell you at a glance whether a fraction simplifies or a division comes out even.
The rules
Factors and multiples. The factors of a number divide into it evenly: the factors of 12 are 1, 2, 3, 4, 6, and 12. The multiples of a number are what you get by multiplying it by 1, 2, 3, and so on: the multiples of 12 are 12, 24, 36, and on forever. Factors are smaller than or equal to the number; multiples are larger than or equal to it.
Divisibility rules. A number is divisible by 2 if it ends in an even digit; by 3 if its digits add to a multiple of 3; by 4 if its last two digits are divisible by 4; by 5 if it ends in 0 or 5; by 6 if it passes both the 2 and 3 tests; by 9 if its digits add to a multiple of 9; by 10 if it ends in 0. So 738 is divisible by 2, by 3, since 7 plus 3 plus 8 is 18, and by 6, but not by 4, since 38 is not, and not by 9, since 18 is not a multiple of 9.
Prime numbers. A prime has exactly two factors, itself and 1. The primes under 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Two is the only even prime. One is not prime. A number that is not prime and not 1 is composite. To check whether a number under 100 is prime, test whether 2, 3, 5, or 7 divides it; if none does, it is prime.
Prime factorization. Every composite number breaks into primes in exactly one way. Split it into any two factors and keep splitting until only primes remain: 60 is 6 times 10, which is 2 times 3 times 2 times 5. Written in order, 60 is 2 times 2 times 3 times 5.
Greatest common factor. The largest number that divides both. For small numbers, list the factors. For larger ones, use prime factorization and multiply the shared primes. The GCF of 18 and 30: 18 is 2 times 3 times 3 and 30 is 2 times 3 times 5, they share one 2 and one 3, so the GCF is 6. Word problems that ask for the largest equal groups or the biggest square tile that fits want the GCF.
Least common multiple. The smallest number that both divide into. For small numbers, list multiples of the larger one until the smaller one divides it. For 6 and 8: 8, 16, 24, and 6 divides 24, so the LCM is 24. Word problems about things happening together again, and fraction sums that need a common denominator, want the LCM.
Worked examples
Example 1
Which of the following is a prime number?
- A.21
- B.27
- C.31
- D.33
Show answer
Answer: C: 31
Twenty-one is 3 times 7, 27 is 3 times 9, and 33 is 3 times 11. Thirty-one is not divisible by 2, 3, 5, or 7, so it is prime. The digit-sum test rules out 21, 27, and 33 quickly, since each has digits adding to a multiple of 3.
Example 2
Which number is divisible by both 3 and 4?
- A.126
- B.232
- C.316
- D.348
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Answer: D: 348
For 3, the digits must add to a multiple of 3; for 4, the last two digits must be divisible by 4. Only 348 passes both: 3 plus 4 plus 8 is 15, and 48 is divisible by 4. The number 126 passes 3 but not 4, and 232 and 316 pass 4 but not 3.
Example 3
What is the prime factorization of 72?
- A.2 × 36
- B.2² × 3²
- C.2³ × 3²
- D.8 × 9
Show answer
Answer: C: 2³ × 3²
Seventy-two is 8 times 9, and 8 is 2 times 2 times 2 while 9 is 3 times 3. That is three 2s and two 3s, written 2 cubed times 3 squared. Choices A and D are factor pairs but not prime factorizations, since 36, 8, and 9 are not prime.
Example 4
What is the greatest common factor of 24 and 40?
- A.4
- B.8
- C.12
- D.120
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Answer: B: 8
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40. The largest shared factor is 8. Choice D, 120, is the least common multiple, a different question.
Example 5
What is the least common multiple of 9 and 12?
- A.3
- B.24
- C.36
- D.108
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Answer: C: 36
Multiples of 12: 12, 24, 36. Nine divides 36, so the LCM is 36. Choice D, 108, is 9 times 12, which is a common multiple but not the least; choice A, 3, is the greatest common factor.
Example 6
A florist has 36 roses and 48 tulips and wants to make identical bouquets with no flowers left over. What is the greatest number of bouquets she can make?
- A.6
- B.12
- C.24
- D.84
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Answer: B: 12
Identical bouquets with nothing left over means the number of bouquets must divide both 36 and 48, and the greatest such number is the GCF, 12. Each bouquet then has 3 roses and 4 tulips.
Example 7
One bus leaves the station every 10 minutes and another every 15 minutes. If both leave at 8:00, when will they next leave together?
- A.8:05
- B.8:25
- C.8:30
- D.9:00
Show answer
Answer: C: 8:30
The buses leave together at every common multiple of 10 and 15 minutes. The least common multiple is 30, so the next time is 30 minutes after 8:00, which is 8:30. Choice D is also a common multiple but not the first.
Common mistakes
- Calling 1 a prime. It has one factor, not two.
- Swapping GCF and LCM. The GCF is small and divides both numbers; the LCM is large and both numbers divide it. Equal groups want the GCF; happening together wants the LCM.
- Stopping the factorization at a composite. Eight times 9 is not a prime factorization; keep splitting.
- Testing divisibility by 4 on the whole number. Only the last two digits matter.
- Assuming odd numbers are prime. Twenty-one, 27, 33, 39, and 51 are all composite.
Practice
These facts appear in arithmetic, in word problems about groups and schedules, and inside number series built on multiples. The fractions lesson uses the GCF to simplify and the LCM to find common denominators.