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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
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B
1683
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C
1697
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D
1703
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Solution

## The correct option is B 1683Let 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+3For different values of k, we get different values of x.k=1⇒x=843, is indivisible by 9k=2⇒x=1683, is divisible by 9∴1683 is the numberwhich when divided by 5, 6, 7 or 8 leaves the remainder 3 in each case,but when divided by 9 leaves no remainder.

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