1.) Prove that a square of some integer has to leave a remainder of either 0 or 1 when it is divided by 4.

2.) Prove that there is no number in the following sequence: 11, 111, 1111, 11111, ... that's a perfect square.

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- February 19th 2007, 10:32 PMIdeasman(Perfect) Squares
1.) Prove that a square of some integer has to leave a remainder of either 0 or 1 when it is divided by 4.

2.) Prove that there is no number in the following sequence: 11, 111, 1111, 11111, ... that's a perfect square. - February 20th 2007, 12:06 AMAlvinCY
1. Let the number you're after be p. There are only 4 possible senarios in modulo 4.

p = 0 (mod 4) => p^2 = 0 (mod 4)

p = 1 (mod 4) => p^2 = 1 (mod 4)

p = 2 (mod 4) => p^2 = 4 = 0 (mod 4)

p = 3 (mod 4) => p^2 = 9 = 1 (mod 4)

Therefore any square number would have the property that p = 0 or 1 (mod 4)

2. As you can see, you can write each of the 1111.....11 (n digits) numbers as:

(n-2) digits ---> 111...111 x 100 + 11

In mod 4, this is the same as 111...111 x (4 x 25) + 11 = 11 = 3 (mod 4)

Since 111...111 is not equal to 0 or 1 mod 4, it cannot be a perfect square (from 1) - February 20th 2007, 12:36 PMThePerfectHacker
This one is simple.

Because,

111..1111 = 111....108 +3

But 111....108 is divisible by 4 for the last two digits are.

Thus, it is expressible as 4k.

Thus,

111....1111=4k+3 for some k in Z.

But by the previous discussion squares are of the form 4k or 4k+1. :eek: Impossible.