# Homework Help: Determine the remainders when dividing their squares by four

1. Apr 29, 2007

### ND3G

Randomly select eight odd integers of less than 1000

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

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?

2. Apr 29, 2007

### 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?

3. Apr 29, 2007

### ND3G

EDIT:

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

The remainder is 1/4

Last edited: Apr 29, 2007
4. Apr 29, 2007

### 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.

5. Apr 29, 2007

### 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

Last edited: Apr 29, 2007
6. Apr 30, 2007

### HallsofIvy

No, the remainder is NOT 1/4!!

Remember what you said initially: