Latest MBA Aptitude Question SOLUTION: The power of 45 that will exactly divide 123! is? kindly post solution as well

1) 28
2) 30
3) 31
4) 59
5) 29

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The power of 45 that will exactly divide 123! is? kindly post solution as well

1) 28
2) 30
3) 31
4) 59
5) 29

Asked In MBA (11 years ago)

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Dr.Dipin Singh  (11 years ago)

28
45= 5*9=5*3^2
123! has 123/5 +123/25 = 24+4=28 multiples of 5.
also 123 has 123/3 +123/9 +127/27+127/81= 41+13+4+1=59 multiple of 3 or 29 multiples of 9.
so 45^28 will exactly divide 123!

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sc mandota  (11 years ago)

28
45= 5*9=5*3^2
123! has 123/5 +123/25 = 24+4=28 multiples of 5.
also 123 has 123/3 +123/9 +127/27+127/81= 41+13+4+1=59 multiple of 3 or 29 multiples of 9.
so 45^28 will exactly divide 123!

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Michael Angelo S. Cenas  (3 years ago)

Prime factorization of 45
45 = 9•5
45 = 3²•5


So, we need to find the highest power of 3 and 5 that divides 123! exactly.

The highest power of 3 that divides 123! is
= ⌊123/3⌋ + ⌊123/3²⌋ + ⌊123/3³⌋ + ⌊123/3⁴⌋, where ⌊x⌋ denotes floor function of x
= 41+13+4+1
= 59

The highest power of 5 that divides 123! is
= ⌊123/5⌋ + ⌊123/5²⌋
= 24+4
= 28

So 3⁵⁹ divides 123! exactly and 5²⁸ divides 123! exactly.

45 is 9•5, we need 3⁵⁹ to change it into power of 9.
3⁵⁹ = 3•(3²)²⁹
3⁵⁹ = 3•9²⁹, the highes power of 9 that divides 123! is 29.

Now, since the power of 5 is lesser than the power of 9. We take the highest power of 5.

Therefore, the highest power of 45 that divides 123! is 28.

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