Problem 17 · 2014 Math Kangaroo
Stretch
Number Theory
divisibility
Lea plays with her marbles, placing them in small groups on the table. In groups of three, two marbles are left over. In groups of five, again two are left over. How many more marbles does Lea need so that she can place them in groups of three and in groups of five with none left over?
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Answer: E — 13
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Hint 1 of 2
Leaving 2 over for both groups of three and groups of five means leaving 2 over for groups of fifteen.
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Hint 2 of 2
She wants a multiple of 15; find the next multiple of 15 above her current count and see how many more marbles that needs.
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Approach: use that 2 left over mod 3 and mod 5 means 2 left over mod 15
- Leaving 2 over in groups of three and in groups of five means leaving 2 over in groups of fifteen.
- So her smallest possible count is 2; to split evenly into both 3s and 5s she needs a multiple of 15.
- The next multiple of 15 after 2 is 15, which is 15 − 2 = 13 more marbles.
- She needs 13 more marbles.
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