Cece's Flower Arrangements

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

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