Question

In congruence modulo $5,\{x\in Z:x=5k+2,k\in Z\}$ represents:

• A. $\left[0\right]$
• B. $\left[5\right]$
• C. $\left[7\right]$
• D. $\left[2\right]$

Answer 1

Answer: (D)

$\left[2\right]$

Lets enumerate all possibe values of x ignoring the modulo part for now.

$x=5k+2$ where k belongs to all integers

Lets start putting $k=0,1,2,...$

when $k=0,x=5.0+2=2$

when $k=1,x=5.1+2=7$

when $k=2,x=5.2+2=12$

when $k=3,x=5.3+2=17$

when $k=4,x=5.4+2=22$

Possible values of $x=\{2,7,12,17,\mathrm{22..}\}$

Now lets apply modulo $5$ to all values.

$x=\{2,2,2,2,\mathrm{2....}\}$

$x$ left with only $2$

Reason: This happens because each number is just 2 more than a multiple of 5.

Thus on taking modulo we are left with only remainder that is 2.