Question

When a number is divided by \(4\), the remainder is \(2\). When the quotient is divided by \(2\), the remainder is \(1\). What will be the remainder when the number is divided by \(8\)?

• A. \(4\)
• B. \(5\)
• C. \(6\)
• D. \(7\)

Hint:

Write \(N=4q+2\) and \(q=2m+1\), then substitute to get \(N=8m+6\).

Answer 1

The Correct Option is C

Step 1: Write the first division as an equation.
Let the number be \(N\) and let \(q\) be the quotient when \(N\) is divided by \(4\). Since the remainder is \(2\):
\[ N = 4q + 2 \]

Step 2: Write the second condition as an equation.
We are told that when this quotient \(q\) is divided by \(2\), the remainder is \(1\). So \(q\) is odd, and we can write
\[ q = 2m + 1 \]
for some whole number \(m\).

Step 3: Substitute back to express \(N\) in terms of \(m\).
\[ N = 4(2m+1) + 2 = 8m + 4 + 2 = 8m + 6 \]

Step 4: Read off the remainder on division by 8.
Since \(N = 8m + 6\), dividing \(N\) by \(8\) always gives quotient \(m\) and remainder \(6\), no matter what whole number \(m\) is, because \(6\) is already less than \(8\).

Step 5: Why the other options are wrong.
Option (a), 4, would be right only if the first condition alone were used and the second condition ignored. Option (b), 5, and option (d), 7, do not come out of \(8m+6\) for any integer \(m\); they would only appear if the two remainder conditions were combined incorrectly instead of substituting properly.

Final Answer:
The remainder when the number is divided by 8 is 6. \[ \boxed{6} \]