Question

The number of divisors of 8400 except unity is

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Answer 1

Answer:

2,3,5,7 are prime, so they have only two divisors. 4 and 25 are squares of primes and have three divisors. All the rest have four or more. There are a total of 5⋅2⋅3⋅2=60 divisors, including the ones above......

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Answer 2

8400=4×100×21=18×25×3×7
=3
1
.2
4
.5
2
.7
1
=3
a
2
b
5
c
7
d

a varies from 0 to 1, b from 0 to 4, c from 0 to 2 and d again from 0 to 1.
Thus the total number of divisors is
= 2×5×3×2=60
We have to exclude the divisor 1 and the divisor 8400. i.e. the number itself. Hence required number is 60−2=58.
Sum of the divisors
Any divisor is of the form 3
a
2
b
5
c
7
d

Sum = ∑
0
1
​
3
a
∑
0
4
​
2
b
∑
0
2
​
5
c
∑
0
1
​
7
d

=(1+3)(1+2+2
2
+2
3
+2
4
)(1+5+5
2
)(1+7)
=4×31×31×8