Question

The least number which when divided by 5, 6 , 7 and 8 leaves a remainder 3, but when divided by 9 leaves no remainder, is:

A : 3168
B : 3138
C : 8163
D : 1683​

Answer 1

The least number which when divided by 5, 6 , 7 and 8 leaves a remainder 3.

→ LCM of 5 = 1 × 5

→ LCM of 6 = 2 × 3

→ LCM of 7 = 1 × 7

→ LCM of 8 = 2 × 2 × 2

So,

→ LCM of 5, 6, 7 and 8 = 2 × 2 × 2 × 3 × 5 × 7

→ 840

As, it leaves a remainder 3. So, it is in the form :

→ 840n + 3

If n = 1. Then,

→ 840(1) + 3

→ 843

Which is not divisible by 9 and leaves a remainder 6.

If n = 2. Then,

→ 840(2) + 3

→ 1680 + 3

→ 1683

Which is divisible by 9 and leaves no remainder.

Answer :

Option D) 1683

Answer 2

$\bold{\underline{\underline{Answer:}}}$

The least number = 1683.

[Option : D]

$\bold{\underline{\underline{Step\:by\:step\:explanation:}}}$

Given :

• Least number which when divided by 5, 6 , 7 and 8 leaves a remainder 3
• When the number is divided by 9 it leaves no remainder.

To find :

• The least number.

Solution :

First finding the LCM of 5,6,7 and 8.

LCM of 5 :-

$\rightarrow$ $\bold{5=1\times\:5}$

LCM of 6 :-

$\rightarrow$ $\bold{6=2\times\:3}$

LCM of 7 :-

$\rightarrow$ $\bold{7=1\times\:7}$

LCM of 8 :-

$\rightarrow$ $\bold{8=2\times\:4}$

$\rightarrow$ $\bold{8=2\times\:2\times\:2}$

LCM of 5,6,7 and 8 :-

$\rightarrow$ 2 × 2 × 2 × 3 × 5 × 7

$\rightarrow$ 840.

Given that, when divided the remainder is 3, making it of the form :-

• 840n + 3

Since we have the option given in the question, we will assume various value of n and substitute it for n in the 840n + 3.

When n = 1 ,

$\rightarrow$ 840(1) + 3

$\rightarrow$ 840 + 3

$\rightarrow$ 843

Given in the question :

• The number when divided by 9 leaves no remainder.

But when we divide 843 by 9,

Remainder = 6

Which means n = 1 doesn't satisfies the condition.

When n = 2 ,

$\rightarrow$ 840n + 3

$\rightarrow$ 840(2) + 3

$\rightarrow$ 1680 + 3

$\rightarrow$ = 1683.

When we divide 1683 by 9, it is completely divisible and leaves no remainder.

•°• The least number which when divided by 5, 6 , 7 and 8 leaves a remainder 3, but when divided by 9 leaves no remainder, is: $\bold{\sf{\large{\green{1683}}}}$