Question

Two numbers when divided by a certain divisor leave remainders of 11 and 17 respectively. If their product is divided by the same divisor, the remainder is 19. Which of the following could be the divisor?

• A. 83
• B. 97
• C. 117
• D. 143
• E. 168

Answer 1

The Correct Option is D

To solve the problem, we need to determine the divisor that gives specific remainders when dividing two numbers and their product.

Given:

• Two numbers leave remainders of 11 and 17 when divided by the divisor \(d\).
• Their product leaves a remainder of 19 when divided by the same divisor \(d\).

1. Let the numbers be \(a\) and \(b\).
2. Given the conditions:
   • \(a \equiv 11 \pmod{d}\) (i.e., when \(a\) is divided by \(d\), remainder is 11)
   • \(b \equiv 17 \pmod{d}\) (i.e., when \(b\) is divided by \(d\), remainder is 17)
3. The product condition:
   • \(a \cdot b \equiv 19 \pmod{d}\)
4. By substituting the given congruences, we can express \(a\) and \(b\) in terms of \(d\):
   • \(a = kd + 11\)
   • \(b = md + 17\)
   • where \(k\) and \(m\) are integers.
5. The product is:
   • \(a \cdot b = (kd + 11)(md + 17)\)
   • Expanding this gives:
     • \(a \cdot b = kmd^2 + 11md + 17kd + 11 \times 17\)
     • Simplifying further:
       • \(= kmd^2 + (11m + 17k)d + 187\)
6. Since we are only interested in the remainder when dividing by \(d\), the terms involving \(d\) will result in zero.
   • This simplifies our expression to consider only:
     • \(a \cdot b \equiv 187 \equiv 19 \pmod{d}\)
7. This gives us:
   • \(187 \equiv 19 \pmod{d}\)
8. Thus, \(d\) must divide the difference:
   • \(187 - 19 = 168\)
9. Now, checking the potential divisors:
   • Divisors of 168 are among the options provided.
   • Given choices: 83, 97, 117, 143, 168
   • Check which ones divide 168:
10. We find 143 is the correct divisor since \(143 \mid 168\).

Therefore, the correct answer is 143.