A number when divided by 6 leaves
remainder 3. When the square of the same
number is divided by 8, the remainder is? plz explain
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If a number when divided by 6 leaves remainder 3, then the number is let as 6n + 3.
Squaring it, we get,
(6n + 3)² = (6n)² + 2(6n)3 + 3²
= 36n² + 36n + 9
= 36n(n + 1) + 9
= 36n(n + 1) + 8 + 1
Here, n(n + 1) is the product of two consecutive natural numbers. That always must be an even number (There has been no doubt for that!).
∴ 36n(n + 1) will be a multiple of 36 x 2 = 72 = 8 x 9.
∴ 36n(n + 1) will also be a multiple of 8.
Adding 8 to 36n(n + 1) also results a multiple of 8. But 1 remains there.
∴The square of the number, which when divided by 6 leaves remainder 3, leaves remainder 1 on division by 8.
But, a funny thing for this is, there is no need for an explanation to find the answer for such questions!
Just find the remainder of (6 + 3)² ÷ 8 {(Divisor by which the number is divided + Remainder got by the division)² ÷ (Divisor by which the square of the number is divided)}. We'll get the answer.
The answer is 1.
Hope this will be helpful.
Squaring it, we get,
(6n + 3)² = (6n)² + 2(6n)3 + 3²
= 36n² + 36n + 9
= 36n(n + 1) + 9
= 36n(n + 1) + 8 + 1
Here, n(n + 1) is the product of two consecutive natural numbers. That always must be an even number (There has been no doubt for that!).
∴ 36n(n + 1) will be a multiple of 36 x 2 = 72 = 8 x 9.
∴ 36n(n + 1) will also be a multiple of 8.
Adding 8 to 36n(n + 1) also results a multiple of 8. But 1 remains there.
∴The square of the number, which when divided by 6 leaves remainder 3, leaves remainder 1 on division by 8.
But, a funny thing for this is, there is no need for an explanation to find the answer for such questions!
Just find the remainder of (6 + 3)² ÷ 8 {(Divisor by which the number is divided + Remainder got by the division)² ÷ (Divisor by which the square of the number is divided)}. We'll get the answer.
The answer is 1.
Hope this will be helpful.
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