# Help ...

What is the remainder if $1^{2} + 3^{2} + 5^{2} + \cdots+ 2011^{2}$ is divided by 8?

There must be an easy way to solve this. But I couldn't remember that! Help me to find the correct path ...

Note by Raiyun Razeen
2 years, 6 months ago

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Although I have provided a solution before, I discovered a much quicker way to do it. If you put them into pairs, it will be

$(1^2+3^2)+(5^2+7^2)+(9^2+11^2)+\dots+(2009^2+2011^2)$

This may be written into the form of

$\sum_{n=1}^{503}((4n-3)^2+(4n-1)^2) = \sum_{n=1}^{503}(32n^2-32n+10)$

Since $$8|32$$, all it matters to find the reminder is

$\sum_{n=1}^{503}10 = 5030 = 8(628)+6$

Hence, the reminder is 6

- 2 years, 6 months ago

Nice pairing.

I'm not sure what your central equation is about, since neither of the expressions depend on $$n$$. Can you edit it accordingly?

Staff - 2 years, 6 months ago

Ouch, thanks Calvin. It supposed to be $$n$$ but I am too familiar with $$x$$ when writing the expression. I'll edit them now.

- 2 years, 6 months ago

I still believe that the equation is wrong. Notice that the RHS is summing up a constant, and doesn't depend on $$n$$. Is this intentional?

Staff - 2 years, 6 months ago

I suppose to write it with $$n$$ by not substituting 503. This is then fixed.

- 2 years, 5 months ago

Looks good now :) Thanks!

Staff - 2 years, 5 months ago

If you apply remainder theorem then remainder of 1^2 , 3^2 , 5^2 , 7^2 .............2011^2 when divided by 8 seperately will all equal to 1. so their will be 1006 times 1 .which totals 1006. Then remainder of 1006/8 =6. So the remainder will be 6.

- 2 years, 6 months ago