Number Properties 3rd Chapt. 2 p.62 #17

[dahart]

If k = 2n - 1, where n is an integer, what is the remainder of k^2 / 8?

I understand this problem except that when I test n with a 1 it doesn't give the answer . . . help, please?

[esledge]

When n = 1, k = 1. (k = 2(1)-1 = 2-1 = 1)

Thus, the remainder of (k^2)/8 is the remainder when 1 is divided by 8. 8 goes evenly into zero (zero times), with 1 left over. The remainder is 1.

A related rule (and frequent question from students) concerns zero. Zero is divisible by any non-zero integer, thus:

The remainder when 11 is divided by 21 is 11. (because 11 = 0*21 + 11)
The remainder when 6 is divided by 10 is 6. (because 6 = 0*10 + 6)
The remainder when 3 is divided by 4 is 3. (because 3 = 0*4 + 3)

Generalizing:
The remainder when x is divided by y > x is x.