To solve the problem, we need to find four consecutive integers whose product is 5040 and then identify the smallest one.

Step 1: Prime factorization of 5040

First, factorize 5040 into its prime components:
[5040 = 2^4 \times 3^2 \times 5 \times 7]

Step 2: Identify consecutive integers

Notice that 7 is a prime factor. Since 14 (the next multiple of 7) is too large (14×13×12×11 > 5040), 7 must be one of the four consecutive numbers.

Check the sequence including 7:
[7 \times 8 \times 9 \times 10 = 5040]

This product exactly equals 5040.

Step 3: Determine the smallest age

The four consecutive ages are 7, 8, 9, and 10. The youngest is 7.

Answer: 7.

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