Question

The least number which when divided by $5,6.7$ or $8$ leaves the remainder $3$ in each case, but when divided by 9 leaves no remainder, is

• A. $1677$
• B. $1683$
• C. $1697$
• D. $1703$

Answer 1

The correct option is B $1683$

Let the number be x.

When x is divided by 5, 6, 7 or 8, it leaves remainder 3.

$\therefore x$ must be in the form of $(k\times y+3),$

where, k is a constant $\Rightarrow k=1,2,\mathrm{3...}$

y is the Least Common multiple (LCM) of 5, 6, 7 and 8

$\therefore y=LCM(5,6,7,8)$

$\Rightarrow y=LCM\left(LCM\right(5,6),LCM(7,8\left)\right)$

$\Rightarrow y=LCM(30,56)$

$\Rightarrow y=840$

$\therefore x=840\times k+3$

For different values of k, we get different values of x.

$k=1\Rightarrow x=843,$ is indivisible by 9

$k=2\Rightarrow x=1683,$ is divisible by 9

$\therefore 1683$ is the number

which when divided by 5, 6, 7 or 8 leaves the remainder 3 in each case,

but when divided by 9 leaves no remainder.