How many divisors (including itself and 1 ) does 8 6 4 0 0 have?
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(7+1) + (3+1) + (2+1) = 15? Why are the divisors being multiplied?
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Good question! every divisor can be written be form 2 7 ∗ 3 3 ∗ 5 2 so every divisor can be arranged as ⇒ d 2 7 ∗ 3 3 ∗ 5 2 = n = 2 p ∗ 3 q ∗ 5 r 2 7 ∗ 3 3 ∗ 5 2
thus, 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 )
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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8 6 4 0 0 = 2 7 ∗ 3 3 ∗ 5 2 No. Of divisors= ( 7 + 1 ) ∗ ( 3 + 1 ) ∗ ( 2 + 1 ) = 9 6