Easy

How many divisors (including itself and 1 1 ) does 86400 86400 have?


The answer is 96.

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2 solutions

Arya Haldar
Aug 27, 2014

86400 = 2 7 3 3 5 2 86400=2^7*3^3*5^2 No. Of divisors= ( 7 + 1 ) ( 3 + 1 ) ( 2 + 1 ) = 96 (7+1)*(3+1)*(2+1) = 96

(7+1) + (3+1) + (2+1) = 15? Why are the divisors being multiplied?

Therverson Kanavathy - 6 years, 9 months ago

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Good question! every divisor can be written be form 2 7 3 3 5 2 2^7*3^3*5^2 so every divisor can be arranged as 2 7 3 3 5 2 d = n = 2 7 3 3 5 2 2 p 3 q 5 r \Rightarrow \frac{2^7*3^3*5^2}{d} = n=\frac{2^7*3^3*5^2}{2^p*3^q*5^r}

thus, d = 2 p 3 q 5 r d=2^p*3^q*5^r Remember that any nonzero number raised to the zeroth power equals 1. So p, q and r can also be 0. So p can be 0, 1, 2, 3, 4, 5, 6 or 7 which gives for 8 possibilities for p. q can be 0,1,2 or 3, which gives 4 possibilities for q.Same for r. The number of possibilities is just one larger than the corresponding exponent.Thus,every power(th) term must be added then multiplied. ( 8 ) ( 4 ) ( 3 ) \rightarrow (8)(4)(3)

Arya Haldar - 6 years, 9 months ago
Ritam Baidya
Dec 4, 2014

86400 = 2 7 x 3 3 x 5*2 ....no. of divisors are (7+1)(3+1)(2+1)= (8)(4)(3) = 96

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