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# Number Theory

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If $n = 2^{10} \cdot 3^{8} \cdot 5^{14}$, how many of the natural-number factors of n are multiples of 150?

Jul 7, 2022

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2^10 x 3^14 x 5^8 =

2^1 x 3^1 x 5^2 =150

Subtract the exponents of the same base numbers and add 1 in each case as follows:

[10 - 1 + 1] x [14 - 1 + 1] x [8 - 2 + 1] =10 x 14 x 7 = 980 such numbers.