Find the least number which when divided by $35,56$ and $91$ leaves the same remainder $7$ in each case.

The smallest number which when divided by $35,56$ and $91=$ LCM of $35,56$ and $91$

$35=5\times 7$

$56=2\times 2\times 2\times 7$

$91=7\times 13$

LCM $=7\times 5\times 2\times 2\times 2\times 13=3640$

The smallest number that when divided by $35,56,91$ leaves a remainder $7$ in each case $=3640+7=3647$.