Told by Integer

Algebra Level 2

"My name is $n$ , I am an integer number. If you subtract $5$ from me, I will be evenly divisible by $6$ .

If you subtract $50$ from me, I will be evenly divisible by $51$ .

If you subtract $55$ from me, I will be evenly divisible by $56$ .

Actually, if you subtract $m$ (any integer) from me, I will be evenly divisible by $m+1$ ."

Assuming all the above is true, evaluate $n$ .

1 solution

Bilel Mabrouk
Jan 11, 2015

n-m is evenly divisible by m+1 so :

$n-m\equiv 0\left[ m+1 \right] \\ n\equiv m\left[ m+1 \right] \\ n\equiv -1\left[ m+1 \right]$

so :

$n=(m+1)\times k-1$ ; where k is an integer

and since m is variable so $n=-1$ is valid no matter what m is