Cece received at least roses from an admirer. When she tried to place roses in each vase, she found that she had roses left over. When she tried to place roses in each vase, she found that she had 2 roses left over. What is the smallest number of roses she could have received?
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Solution 1: We seek the smallest integer solution to x ≡ 3 ( m o d 5 ) , x ≡ 2 ( m o d 6 ) and x ≥ 1 0 0 . Combining the first 2 equations by the Chinese Remainder Theorem, we get that x ≡ 8 ( m o d 3 0 ) . Hence, x = 8 , 3 8 , 6 8 , 9 8 , 1 2 8 … , so the smallest number of roses Cece could have received is 1 2 8 .
Solution 2: When placing 5 roses in each vase, she had 3 left over. This means that she could have received 1 0 3 , 1 1 3 , 1 1 8 , 1 2 3 , 1 2 8 , … roses. When placing 6 roses in each vase, she had 2 roses left over. This means that she could have received 1 0 4 , 1 1 0 , 1 1 6 , 1 2 2 , 1 2 8 , … roses. The answer is the smallest number than appears in both sequences. Hence, the smallest number of roses she could have received is 1 2 8 .