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# Find the least positive four-digit solution $r$ of the congruence $r^2 + 4r + 4 \equiv r^2 + 2r + 1 \pmod{55}$.

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Find the least positive four-digit solution r of the congruence $$r^2 + 4r + 4 \equiv r^2 + 2r + 1 \pmod{55}$$.

Jan 27, 2018

#1
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Are you sure your congruence is written correctly? With a modulus of 55, there doesn't appear to be a solution !!. It can be written as follows: (r + 2)^2 ≡(r + 1)^2 mod 55.

Jan 27, 2018
#2
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Find the least positive four-digit solution r of the congruence .

$$r^2 + 4r + 4 \equiv r^2 + 2r + 1 \pmod{55}$$

It can be written as follows: (r + 2)^2 ≡(r + 1)^2 mod 55.  ( just as guest said )

(r + 2)^2 ≡(r + 1)^2 mod 55.

Let x=r+1

$$(x+1)^2\equiv x^2 \pmod{55}\\ x^2+2x+1\equiv x^2 \pmod{55}\\ 2x+1\equiv 0 \pmod{55}\\ 2x+1=55k \qquad k\in Z\\ 2x=55k-1 \qquad k\in Z\\ \text{(x-1) must have 4 digits}\\ \text{ k=37 tis the smallest, k needs to be odd}\\ Try\;\; k=37, 39,41,43,.....\\ 2x+1= 2035, 2145, 2255, 2365,\dots\\ 2x=2034, 2144, 2254, 2364, \dots \\ x=1017,1072, 1127, 1182 \dots\\ r=1016, 1071, 1126, 1181 \dots (1016+55c) \qquad c\in Z$$

check

mod(1017^2,55) = 14

mod(1018^2,55) = 14

mod(1072^2,55) = 14

mod(1073^2,55) = 14

Well the first 2 work anyway , so it looks ok

Jan 28, 2018
edited by Melody  Jan 28, 2018
#3
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Thanks!

RektTheNoob  Jan 29, 2018
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I've done some homework this morning and I do not think it is quite right.

According to what I read this is only one answer. The same answer written in different ways.

Plus

I probably didn't need to make it look so complicated ...

Here is the answer that I found before, only I have done it much more simply.

$$r^2 + 4r + 4 \equiv r^2 + 2r + 1 \pmod{55}\\ 4r + 4 \equiv 2r + 1 \pmod{55}\\ 2r \equiv -3 \pmod{55}\\ 2r \equiv 52 \pmod{55}\\ r \equiv 26 \pmod{55}\\ r \equiv 26+55k \pmod{55}\\ r \equiv 26+1045 \pmod{55}\\ r \equiv 1071 \pmod{55}\\$$

Now i need to find 3 other answers.  (Assuming that if I just add multiples of 55 I will just keep getting the same answer expressed in different way.)

Mmm.....

I don't know about any other ones .... Jan 29, 2018
edited by Melody  Jan 29, 2018