Cece's Flower Arrangements

Cece received at least 100 100 roses from an admirer. When she tried to place 5 5 roses in each vase, she found that she had 3 3 roses left over. When she tried to place 6 6 roses in each vase, she found that she had 2 roses left over. What is the smallest number of roses she could have received?


The answer is 128.

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1 solution

Arron Kau Staff
May 13, 2014

Solution 1: We seek the smallest integer solution to x 3 ( m o d 5 ) x \equiv 3 \pmod{5} , x 2 ( m o d 6 ) x \equiv 2 \pmod{6} and x 100 x \geq 100 . Combining the first 2 equations by the Chinese Remainder Theorem, we get that x 8 ( m o d 30 ) x \equiv 8 \pmod{30} . Hence, x = 8 , 38 , 68 , 98 , 128 x = 8, 38, 68, 98, 128 \ldots , so the smallest number of roses Cece could have received is 128 128 .

Solution 2: When placing 5 5 roses in each vase, she had 3 3 left over. This means that she could have received 103 , 113 , 118 , 123 , 128 , 103, 113, 118, 123, 128, \ldots roses. When placing 6 6 roses in each vase, she had 2 2 roses left over. This means that she could have received 104 , 110 , 116 , 122 , 128 , 104, 110, 116, 122, 128, \ldots roses. The answer is the smallest number than appears in both sequences. Hence, the smallest number of roses she could have received is 128 128 .

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