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# Number Theory-Help

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

Apr 6, 2020

#1
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$$n = 2^{10} \cdot 3^{14} \cdot 5^{8}$$

150=5^2*3*2

We have $$2^9*3^{13}*5^6$$ for each number of exponent, it can be up to that or less. or none. so we get 10*14*7 soo my answer is 980.

i think

Apr 6, 2020
#2
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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 + 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.

Apr 6, 2020