Question

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

Answer 1

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

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 (402) 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.

The question is "what is the last non–zero digit in the base 8 expansion?"

Hence the answer is "4"

Choice C is the correct answer.