Answer to Question #312885 in Combinatorics | Number Theory for Awa

Question

Prove or disprove: For all x ∈Z,if x mod11=5,then x 2 mod11=3.

Answer 1

Solution

Given that

$x$ mod11 $= 5$

Then, for some value of "k", we can write x mod11= 5, as

$x = 11k + 5$

Then

$x^{2}=(11k+5)(11k+5)$

$x^{2}=121k^2+55k+55k+25$

$x^{2}=121k^2+110k+25$

Now,

$x^2$ mod 11$=(121k^2+110k+25)$ mod 11

$x^2$ mod 11$=121 k^2$ mod 11 + $110k$ mod 11+ $25$ mod 11

$x^2$ mod 11$=0$ + 0+ 3 because $25=2\times11+3$

Therefore,

$x^2$ mod 11 $= 3$