8 is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides 12!
We are asked the sum of the exponents of the prime factors of the square root of the largest perfect square that divides 12!.
As, the largest perfect square that divides 12! is
2^10 * 3^4 *5^2 = 2073600.
Now, the square root of the number 2073600 is,
1440
Now, we have to find the prime factorization of 1440
1440 = 2×2×2×2×2×3×3×5
Now, in the prime factorization of 1440
the exponent of 2 is 5, exponent of 3 is 2 and the exponent of 5 is 1.
So, the sum of the exponents of the prime factors be 5+2+1 = 8
Hence, 8 is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides 12!
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