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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?

 

I got 702, but that was wrong...

 Apr 6, 2020
 #1
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I'm not the best at this, but this is my attempt:

 

150  =  2 x 3 x 5 x 5  --  so every multiply must have at least 1 two, 1 three, and 2 fives.

 

So, from the 210 (which consists of 10 twos),

you can select either 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10 twos  -- 10 different selections.

 

From 314 (which consists of 14 threes), 

you can select either 1, 2, 3, ... 14 threes  --  14 different selections.

 

From 58, you must select at least 2 fives, but it could also be 3, 4, 5, 6, 7, or 8 fives

  --  7 different selections.

 

Multiplying 10 x 14 x 7, I get 980 different ways.

 Apr 6, 2020
 #2
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That is correct! Thank you so much, geno!

Guest Apr 6, 2020

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