CAT Practice : Number System: Factorial

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A number in base 8 that ends in 4 is a multiple of 4 but not 8. If you 'get' this idea, this question should be easy.

Factorials - base 8

Q.8: When 40! is expressed in base 8 form, what is the last non–zero digit in the base 8 expansion?
1. 2
2. 6
3. 4
4. 2 or 6

Choice C. 4

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Detailed Solution

We need to find the largest power of 8 that divides 40!.
We need to find the largest power of 2 that divides 40!

This is given by $\left( {{{40} \over 2}} \right)$ and then successive division by 2. = 20 + 10 + 5 + 2 + 1 = 38

So, 238 divides 40! Or, (23)12 × 22 divides 40!

(23)12 divides the number, or the base 8 representation ends with 12 zeroes. Now, the base 8 representation of this number will be some (abcd…n)8 × (1000000000000)8. Now, (abcd…n)8 does not end in 0 and is a multiple of 22. The last digit has to be 4.

The last non–zero digit is 4.

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