Question

When a natural number x is divided by 5, the remainder is 2. When a natural number y is divided by 5, the remainder is 4. The remainder is z when x+y is divided by 5. The value of $\frac{2z-5}{3}$ is

• A. -1
• B. 1
• C. -2
• D. 2

Answer 1

The correct option is A -1

According to Euclid's Division Lemma,If $a$ and $b$ are two positive integers such that

$"a"$ is divided by $"b"$ then there exists two uniques integers $"m"$ and $"n"$ such that

$a=bm+n,0\le n<b$

It is given that when a natural number $x$ is divided by $5$,the remainder is $2$.

So for an integer $m$, we have

$x=5m+2$

Also when a natural number $y$ is divided by $5$, the remainder is $4$

So for an integer $n,$ we have

$y=5n+4$

$\Rightarrow x+y=5m+2+5n+4=5(m+n)+6=5(m+n+1)+1$

This shows that when $x+y$ is divided by $5$, we get $1$ as a remainder.

Thus, by given condition

$z=1$

So, the value of $\frac{2z-5}{3}$ is given by

$\frac{2z-5}{3}=\frac{2\left(1\right)-5}{3}=\frac{2-5}{3}=-1$