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To solve the problem of finding the smallest number that satisfies the given conditions, we can follow these steps:

Step 1: Combine the conditions for division by 3 and 7

The number leaves a remainder of 2 when divided by both 3 and 7. Thus, the number minus 2 must be divisible by the least common multiple (LCM) of 3 and 7.
LCM(3,7)=21, so the number can be written as:
N = 21k + 2 (where k is a non-negative integer).

Step 2: Apply the condition for division by 5

The number leaves a remainder of 3 when divided by 5. Substitute N=21k+2 into this condition:
21k + 2 ≡ 3 mod 5

Simplify 21 mod5: 21=4×5+1 → 21≡1 mod5.
Thus:
1×k +2 ≡3 mod5 → k+2=3 mod5 → k≡1 mod5.

So k=5m+1 (where m is a non-negative integer).

Step3: Find the minimal N

Substitute k=5m+1 back into N=21k+2:
N=21(5m+1)+2=105m+23.

The smallest value occurs when m=0: N=23.

Answer: 23

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