Determine the remainders when dividing their squares by four

[ND3G]

Randomly select eight odd integers of less than 1000

a) Determine the remainders when dividing their squares by four, and tabualte your results

b) Make a conjecture about your findings

c) test your conjecture with at least five larger integers

d) Prove of justify the conjecture you make.

n tn = n^2 / 4 - 1, Remainder
49 600 1
81 1640 1
121 3660 1
225 12656 1
441 48620 1
625 97656 1
841 176820 1
-----------------------------------------
2001 1001000 1
3673 3372732 1
6925 11988906 1
8123 16495782 1
9999 249950000 1

b) When the square of an odd integer, for n is equal to or greater than 1, is divided by four the result is an even integer with a remainder of 1.

c)
You know that tn = n^2 / 4 - 1, by hypothesis
Factoring n, tn = (n x n)/4 -1

I'm not quite sure where to take my proof from here. Could someone give me a nudge in the right direction?

 

[HallsofIvy]

These are odd integers! Any odd integer can be written in the form
2n+ 1. What is the square of that? what do you get when you divide by 4?

 

[ND3G]

EDIT:

(2n+1)^2 / 4 = n^2 +n + 1/4 = n(n +1) + 1/4

The remainder is 1/4

 

[Mystic998]

(2n + 1)^2 is not 4n^2 + 1.

And I think he was more concerned with the remainder you get when you divide by 4, not the expression in terms of fractions.

 

[ND3G]

Proof

An odd integer is a multiple of 2, plus 1. Let 2n + 1 represent an odd integer.

The square of an odd integer is 4n^2 + 4n +1 which when divided by 4 becomes n^2 + n +1/4.

Factoring n, tn = n(n+1) + 1/4

An odd number multiplied by itself will result in another odd number. adding two odd numbers will result in an even number.

Therefore, for odd integers, n(n +1) + 1/4 is an even number with a remainder of 1/4

 

[HallsofIvy]

No, the remainder is NOT 1/4!

Remember what you said initially:

b) When the square of an odd integer, for n is equal to or greater than 1, is divided by four the result is an even integer with a remainder of 1.