KA-0062Factors, multiples, LCM and GCD by listing
4 points, difficulty 2 of 3Level 5–6about 75s24 pencils and 36 erasers are shared into identical gift bags, using every item. What is the largest number of bags possible?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
The bags must be identical, so the number divides both.
2The strategy
Every bag holds the same number of pencils and the same number of erasers, so the number of bags divides 24 and divides 36. You want the largest number that does both.
3The full solution
The common factors of 24 and 36 are 1, 2, 3, 4, 6 and 12. The largest is 12, giving 2 pencils and 3 erasers per bag.
Solution
The reliable way
Identical bags with nothing left over means the number of bags divides both 24 and 36. List the factors: 24 gives 1, 2, 3, 4, 6, 8, 12, 24 and 36 gives 1, 2, 3, 4, 6, 9, 12, 18, 36. The largest number in both lists is 12, so 12 bags, each with 2 pencils and 3 erasers. The transferable idea: sharing equally with nothing left over is a greatest-common-divisor question, while coinciding cycles are a least-common-multiple one.
The elegant way
24 = 2 x 2 x 2 x 3 and 36 = 2 x 2 x 3 x 3, so the shared part is 2 x 2 x 3 = 12.
Why this is on the test: Choosing between the greatest common divisor and the least common multiple is the whole question, and picking the wrong one gives an answer that looks reasonable.