Question

Find the least pumber which when divided by 5, 6, 7 and 8 leaves a remainder 3, but when divided by 9 leaves no remainder.​

Answer 1

Answer is 1683

This is the method

Let the number be x.

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

∴x must be in the form of (k×y+3),

where, k is a constant ⇒k=1,2,3...

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

∴y=LCM(5,6,7,8)

⇒y=LCM(LCM(5,6),LCM(7,8))

⇒y=LCM(30,56)

⇒y=840

∴x=840×k+3

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

k=1⇒x=843, is indivisible by 9

k=2⇒x=1683, is divisible by 9