SHSAT: Factors, multiples and remainders
Choose between the GCF and the LCM, count divisors from a prime factorization, and solve remainder and divisibility questions.
How do you decide between the greatest common factor and the least common multiple, and how do you handle remainders?
Short answer
Factor into primes. Use the LCM when cycles must line up and the GCF when you split things into identical groups. For remainders, work with the remainder and reduce it below the divisor.
Know first: Prime factorization, Exponents for repeated multiplication
Step 1 of 5
Factors split things up; multiples line things up
Two tools solve most of these questions. The greatest common factor (GCF) is the largest number that divides two numbers evenly. Use it to split things into the most identical groups. The least common multiple (LCM) is the smallest number both divide into. Use it when two cycles line up again.
Prime factorization finds both. For the GCF, take the primes the numbers share, each at its lower power. For the LCM, take every prime that appears, each at its higher power.
| The question asks | Use | Example |
|---|---|---|
| when two repeating events next happen together | LCM | every 12 s and 18 s: 36 s |
| the most identical groups using everything | GCF | 40 pencils, 56 erasers: 8 kits |
| how many positive divisors | add 1 to each exponent, then multiply | 72 = 2³ × 3²: 4 × 3 = 12 |
| the remainder of a multiple of n | work with the remainder of n | n leaves 2 when divided by 5, so 4n leaves 8 − 5 = 3 |
Write each number as a product of primes first.
Step 2 of 5
Factor first, then build the LCM
Worked example: Two blinking lights
Two lights blink every 12 seconds and every 18 seconds. They blink together now. After how many seconds will they next blink together?
- A6
- B30
- C36
- D216
- 1
Factor
12 = 2² × 3 and 18 = 2 × 3².
- 2
Build the LCM
Take each prime at its higher power: 2² × 3² = 36.
- 3
Check
36 ÷ 12 = 3 and 36 ÷ 18 = 2, and no smaller number works for both.
- 4
See why the other choices are there
6 is the GCF, which divides both numbers but is not a time they share. 30 adds the cycles. 216 multiplies them, which is a common multiple but not the least.
Answer
36 seconds
Find the wrong step
An integer n leaves remainder 5 when divided by 8. What remainder does 3n leave when divided by 8?
- 1
Write n = 8k + 5 for some whole number k.
- 2
Multiply: 3n = 24k + 15.
- 3
24k is a multiple of 8, so the remainder is 15.
What went wrong
A remainder must be less than the divisor, 8. The 15 still contains another 8.
The fix
15 = 8 + 7, so the remainder is 7. Check with n = 13: 3 × 13 = 39 = 4 × 8 + 7.
Step 3 of 5
Pick the tool, then check
Check yourself · Question 1
How many positive divisors does 180 have?
Answer: D
180 = 2² × 3² × 5¹. Each exponent can be used from 0 up to its value, so there are (2 + 1)(2 + 1)(1 + 1) = 3 × 3 × 2 = 18 divisors.
Check yourself · Question 2
What is the smallest positive integer that leaves remainder 2 when divided by 3 and remainder 3 when divided by 5?
Answer: B
List numbers with remainder 3 when divided by 5: 3, 8, 13, 18, 23. Check each for remainder 2 when divided by 3: 3 gives 0, and 8 gives 2. So 8 is the smallest.
Check yourself · Question 3
How many integers from 1 through 100 are divisible by 4 or by 10?
Answer: C
Multiples of 4: 25. Multiples of 10: 10. Numbers divisible by both are multiples of 20: 5. Count those once: 25 + 10 − 5 = 30.
Trap answer: 35
Double counting the overlap
- Why it looks right
- Adding the two lists feels like it covers every number divisible by 4 or by 10.
- Why it's wrong
- Numbers such as 20 and 40 are on both lists, so they were counted twice.
- The right answer
- 30: subtract the 5 multiples of 20 once.
Common mistake
Using the GCF when two cycles have to line up.
- Why it's tempting
- The GCF is the first "common" number you learned, and it is small and quick to find.
- Do this instead
- Ask what the answer measures. A time when both events happen together must be a multiple of each cycle, so you need the LCM. The GCF is for splitting into equal groups.
Step 4 of 5
What to remember
Remember
- LCM for cycles that line up again, GCF for the most identical groups.
- Count divisors by adding 1 to each prime exponent and multiplying.
- For remainder questions, work with the remainder itself and reduce until it is less than the divisor.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
In the practice question, how did you know you needed a common multiple, and how did you check that your answer was the smallest one?