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What is the least number that should be multiplied to 100!

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What is the least number that should be multiplied to 100! [#permalink] New post 06 May 2005, 09:05
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What is the least number that should be multiplied to 100! to make it perfectly divisible by 3^50?

(1) 144
(2) 72
(3) 108
(4) 216
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Re: PS - least number [#permalink] New post 06 May 2005, 09:29
christoph wrote:
What is the least number that should be multiplied to 100! to make it perfectly divisible by 3^50?

(1) 144
(2) 72
(3) 108
(4) 216


Let's count 3s in prime factoring of 100!

Multiples of 3 - 33
Among them multiples of 3^2=9 - 11
Among them multiples of 3^3=27 - 3
Among them multiples of 3^4 = 81 - 1

Total 4*1 + 2*3 + 8 * 2 + 22 * 1 = 48

We need at least another 3^2 = 9

72 = 3^2 * 2^3 - the least number that we need.

Answear (2).
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 [#permalink] New post 06 May 2005, 15:22
one multiple of 3: 3,6,9,.......99 (33)

two multiples of 3: 9,18,27,36,.....99 (11)

three multiples of 3: 27,54, 81 (3)

four multiples of 3: 81 (1)

total: 48

Hence, we need 2 additional multiples of 3.

72 is the least
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 [#permalink] New post 06 May 2005, 16:52
The numbers are from 1 to 100

The biggest number is 99. Lets remove the numbers which are factors of 3 and list them after factoring.

99 = 33 * 3
96 = 32 * 3
.
.
3 = 1 * 3

So,
3 3 3 3 3 3 3 3 3..........3 3 3 3 = total 33 times 3 is a factor
1 2 34 5 6.....32...33= total 11 time 3 appears once more (33/3)

....9........18.........27 = total 3 times 3 appears twice
.................... 27 = total 1 time 3 appears thrice

total 48 times it appears.........only 2 more required...ie smallest number having 9 as a factor is 72.
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  [#permalink] 06 May 2005, 16:52
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