In this lesson:
By the end, you will:
A baker has chocolate cookies and vanilla cookies.
She wants to pack identical bags — same number of chocolate, same number of vanilla, nothing left over.
What is the largest number of bags she can make?
A factor of a whole number divides it evenly.
The factors of are .
A common factor of two numbers divides both.
Common factors of and : .
Walk pairs in order; stop when $n \times n > $ the number.
Factors of :
Common factors of and :
The greatest common factor (GCF) is the largest common factor.
Twelve divides both, and no larger number does.
Common factors:
Common factors: only.
Numbers whose only common factor is are relatively prime.
Tempting answer:
Actual answer: ✓
Factors of : — but does not divide .
The GCF is at most the smaller number — not always equal to it.
That is twelve factor pairs to find — accuracy gets fragile.
A faster method uses prime factorization.
Every whole number greater than factors uniquely into primes.
The prime factorization is the multiplicative anatomy of a number.
Common primes at lowest power: and .
A multiple of a number is its product with any whole number.
Multiples of :
The least common multiple (LCM) is the smallest multiple shared by both numbers.
Smallest common multiple: .
For every prime in either factorization, take it at its highest power.
Highest powers: and .
LCM(8, 12): , , so .
LCM(9, 12): , , so .
Highest power of each prime — across both numbers.
State both rules together every time. They are easy to swap if learned alone.
Why? They share a factor of , which is counted only once.
The product works only when the two numbers are coprime.
For each pair, find the GCF and the LCM:
Try both methods on the first one. Use whichever feels faster on the others.
If you wrote , you assumed the smaller number works.
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Find the greatest common factor and the least common multiple