CSET Multiple Subjects Subtest II: Prime Factorization, Worked Step by Step
Here is a CSET Multiple Subjects Subtest II question of the kind that stops people every test window.
Find the greatest common factor and the least common multiple of 84 and 360.
Try it before you read on. Then compare your method to this one.
Step 1 — Break each number into primes
Keep dividing by the smallest prime that fits.
84 ÷ 2 = 42 · 42 ÷ 2 = 21 · 21 ÷ 3 = 7 · 7 is prime.
So 84 = 2 × 2 × 3 × 7, which we write as 2² × 3 × 7.
360 ÷ 2 = 180 · 180 ÷ 2 = 90 · 90 ÷ 2 = 45 · 45 ÷ 3 = 15 · 15 ÷ 3 = 5 · 5 is prime.
So 360 = 2³ × 3² × 5.
Step 2 — GCF: take the primes they share, lowest power
Line them up and look only at what both numbers have.
2: one has 2², the other 2³. Take the lower → 2²
3: one has 3, the other 3². Take the lower → 3
7 is only in 84. 5 is only in 360. Neither counts.
GCF = 2² × 3 = 12
Step 3 — LCM: take every prime, highest power
Now take every prime that appears in either number, at its highest power.
LCM = 2³ × 3² × 5 × 7 = 8 × 9 × 5 × 7 = 2520
Check your answer in five seconds
For any two numbers, GCF × LCM = the product of the two numbers.
12 × 2520 = 30,240. And 84 × 360 = 30,240. They match, so both answers are right.
Use that check on test day. It catches almost every slip.
The mistake that costs the mark
Most candidates swap the two rules. They take the highest power for the GCF and the lowest for the LCM.
Remember it by size. The greatest COMMON factor has to divide into both numbers, so it must be small. The least COMMON multiple has to contain both numbers, so it must be big. If your GCF came out bigger than either number, you have them backwards.
Watch this one worked out
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Why Subtest II catches good people
Subtest II covers mathematics and science together. The math is not hard mathematics — it is middle school content. But it is content most adults have not touched in fifteen or twenty years.
That is the whole problem. You are not learning this. You are remembering it. Those need different kinds of practice, and most study guides are written for learning.
A few hours with someone who finds your exact gaps usually beats weeks of reading a book cover to cover.
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