Question

A number leaves a remainder of 61 when divided by 783. What will be the remainder when the same number is divided by 27?

• A. 8
• B. 5
• C. 7
• D. 6

Answer 1

Answer: (C)

7

Answer 2

Let the number be $N$. According to the division algorithm, when $N$ is divided by 783, the remainder is 61. This can be written as:

$N = 783 \times q + 61$

where $q$ is the quotient and $0 \le 61 < 783$.

We need to find the remainder when the same number $N$ is divided by 27. We observe that the first divisor, 783, is a multiple of the second divisor, 27.

Let's divide 783 by 27:

$783 \div 27 = 29$

So, $783 = 27 \times 29$.

Substitute this into the equation for $N$:

$N = (27 \times 29) \times q + 61$

$N = 27 \times (29q) + 61$

Now, we want to find the remainder when $N$ is divided by 27. The term $27 \times (29q)$ is clearly divisible by 27. Therefore, the remainder when $N$ is divided by 27 will be the same as the remainder when 61 is divided by 27.

Let's divide 61 by 27:

$61 = 27 \times 2 + 7$

Here, the quotient is 2 and the remainder is 7. Since $0 \le 7 < 27$, the remainder is 7.

Substituting this back into the expression for $N$:

$N = 27 \times (29q) + (27 \times 2 + 7)$

$N = 27 \times (29q) + 27 \times 2 + 7$

Factor out 27 from the terms divisible by 27:

$N = 27 \times (29q + 2) + 7$

This equation is in the form $N = 27 \times Q + R$, where $Q = 29q + 2$ is the new quotient and $R = 7$ is the remainder.

Since $0 \le 7 < 27$, the remainder when $N$ is divided by 27 is 7.