A number when divided by 2,3,4,5,6 leaves remainder 1,2,3,4,5 respectively but when its divided by 11 the remainder is 0. FIND THE NUMBER

Question

A number when divided by $2,3,4,5,6$ leaves remainder $1,2,3,4,5$ respectively but when its divided by $11$ the remainder is $0$. FIND THE NUMBER

I tried taking LCM of $2,3,4,5,6$ and subratcing by $1(59)$ then multiply by $11$. What am i doing wrong? Please Guide.

Answer 1

Let $n$ be that number then $n+1$ can divisible by $2,3,4,5,6$ without remainder.

$\newcommand{\lcm}{\operatorname{lcm}}\lcm(2,3,4,5,6)=60$

$n+1=60k, k\in Z$

and if the number is divisible by $11$

$n=11t,t\in Z$

by solving

$$n+1=60k, k\in Z$$ $$n=11t,t\in Z$$

$n=60k-1=11t$, $ k , t\in Z$

$t=5k+\frac{5k-1}{11}$

for $k=9, t=49$

then

$$n=539$$

In general $$n\equiv 539 \mod {660}$$