Question

The least multiple of $13$ which when divided by $4,5,6,7$ leaves remainder $3$ in each case is

• A. $3783$
• B. $3780$
• C. $2520$
• D. $2522$

Answer 1

Answer: (A)

$3783$

$LCM$ of $\{4,5,6,7\}=420$

A number which leaves a remainder of ${}^{\prime }{3}^{\prime }$

When divided by $4,5,6,7$ is of the form :

$N=420K+3$ ; $Kϵ$ integers

Now,

$N=416K+(4K+3)$

$=416K+4K+3$

For ${}^{\prime }{N}^{\prime }$ to be divisible by ${}^{\prime }{13}^{\prime }$ we need to have $4K+3$ divisible by $13$ as $416K$ is always divisible by $13$. So the smallest possible $K$ will be $9$ for which $4K+3=39$

$\therefore$ the smallest / least multiple of $13$ which when divided by $4,5,6,7$ leaves remainder $3$ in each case is

$N=420\times 9+3$

$=3780+3$

$=3783$

Correct answer is option $A$.