Question : Calculate the total number of prime factors in the expression $(4)^{11}×(5)^{5}×(3)^{2}×(13)^{2}$.
Option 1: 30
Option 2: 31
Option 3: 33
Option 4: 32
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Correct Answer: 31
Solution :
Given: $(4)^{11}×(5)^{5}×(3)^{2}×(13)^{2}$
= $(2^{2})^{11}×(5)^{5}×(3)^{2}×(13)^{2}$
= $(2)^{22}×(5)^{5}×(3)^{2}×(13)^{2}$
If $N = a^m × b^n$, where a and b are prime numbers, then the total number of prime factors $N = m + n$
$\therefore$ Total prime factors = 22 + 5 + 2 + 2 = 31
Hence, the correct answer is 31.
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