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: Find the smallest number that fits both conditions.
We need $N$ divided by $4$ to leave remainder $2$, and the resulting quotient divided by $2$ to leave remainder $1$, meaning the quotient must be odd.
Try quotient $q=1$ (the smallest odd number): $N = 4(1)+2 = 6$.
Check: $6 \div 4 = 1$ remainder $2$. Correct. And $q=1$ divided by $2$ gives remainder $1$. Correct.

Step 2: Divide this example number by 8.
$6 \div 8 = 0$ remainder $6$.

Step 3: Test a second example to make sure this is not a coincidence.
Try quotient $q=3$ (the next odd number): $N = 4(3)+2=14$.
Check: $14 \div 4 = 3$ remainder $2$. Correct. And $q=3$ divided by $2$ gives remainder $1$. Correct.
Now $14 \div 8 = 1$ remainder $6$.

Step 4: Try one more to confirm the pattern holds.
Try quotient $q=5$: $N=4(5)+2=22$. Check both conditions hold, and $22 \div 8 = 2$ remainder $6$.
Every valid $N$ lands on remainder $6$ when divided by $8$, because each time the quotient jumps by $2$ (staying odd), $N$ jumps by exactly $8$, which does not change the remainder on division by $8$.

Final Answer:
Testing actual numbers that satisfy both conditions always gives remainder 6 on division by 8. \[ \boxed{6} \]