Lots of factors

Number Theory Level 2

How many factors (prime and composite) inclusive of 1 and itself are there for the number 5 × 1 0 10 5\times 10^{10} ?


The answer is 132.

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Prithu Purkait by prithu purkait , 20, India

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3 solutions

Neeraj Kamal
Sep 13, 2015

5×10^10=5^11×2^10, so total no. of factors of 2^10×5^11=(10+1)×(11+1)=11×12=132

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Huân Lê Quang
Jul 13, 2015

We have: 5 x 10^10 = (2^10) x (5^11)

So count the case:

1 (1 proper divisor, trivially)

From 5 to 5^11 (11 proper divisors)

From 2 to 2^10 (10 proper divisors)

The combinations: 11 x 10 = 110 (proper divisors)

So, number of proper divisors of (5 x 10^10) = 1+11+10+110 = 132

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Sai Karthik K A
Jul 13, 2015

proper prime factorization must be done......... 5^5 X 10^10= 5^11 x 2^10. therefore, no of factors=(11+1)(10+1) =12 x 11=132

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