What is the least number which when divided by 5, 6, 7 and 8 leaves a remainder 3, but when divided by 9 leaves no remainder?
Answers
Least number which when divided by 5,6,7 and 8 leaving remainder 3 =
LCM of 5,6,7 and 8.
(i) Prime factorization of 5 = 5
(ii) Prime factorization of 6 = 2 * 3
(iii) Prime factorization of 7 = 7
(iv) Prime factorization of 8 = 2 * 2 * 2
LCM(5,6,7,8) = 5 * 3 * 2 * 2 * 2 * 7
= 840.
Given that it leaves a remainder 3.
So, the number is of the form = 840k + 3{bq + r}
When k = 1:
= > 840(1) + 3
= > 843
Not divisible by 9 and leaves remainder 6.
When k = 2:
= > 840(2) + 3
= > 1680 + 3
= > 1683.
Divisible by 9 and leaves no remainder.
Therefore, the least number is 1683.
Hope it helps!
The least number is 1683
Explanation:
Given Condition:
1. The least number which when divided by 5, 6, 7 and 8 leaves a remainder
of 3
2. But when divided by 9 leaves no remainder
To find:
The least number
Solution:
==> LCM of 5,6,7 and 8
==> Here, 5 and 7 are prime numbers
==> The LCM of 6 and 8 = 24
==> LCM of 5,6,7 and 8 = 5×7×24
==> LCM of 5,6,7 and 8 = 840
==> The number be x
==> Number = 840x
==>The Remainder is added to the number.
==> Number = 840x +3
==> Assume the value x = 1
==> Least Number = 840×1 +3
==> Least Number = 840 +3
==> Least Number = 843
==> 843 is not divisible by 9
==> Assume x=2
==> Least Number = 840×2 +3
==> Least Number = 1680 +3
==> Least Number = 1683
==> 1683 is divisible by 9
==> Which leaves no remainder
==> The least number is 1683
==> Check it by dividing the number by 5,6,7 and 8
==> 1683÷5
==> Quotient is 1680
==> Remainder = 3
==> 1683÷6
==> Quotient is 1680
==> Remainder = 3
==> 1683÷7
==> Quotient is 1680
==> Remainder = 3
==> 1683÷8
==> Quotient is 1680
==> Remainder = 3