Question

Which of the following integers is congruent to 5 module 17 .

I. 80
II. 18
III. -29
IV. -3
(a) II and IV
(b) I
(c) III
(d) III and IV
(e) none of them

Answer 1

Upon testing each number provided, it is revealed that only -29 is congruent to 5 modulo 17. Therefore, the correct answer to the student's question is (c) III.

Step-by-step explanation:

The question is asking which of the following integers is congruent to 5 modulo 17. In modular arithmetic, two numbers are congruent modulo 17 if they have the same remainder when divided by 17. We can test each provided number to see if it is congruent to 5 modulo 17.

• Option I: 80 ÷ 17 gives a remainder of 12, so 80 is not congruent to 5 modulo 17.
• Option II: 18 ÷ 17 gives a remainder of 1, so 18 is not congruent to 5 modulo 17.
• Option III: -29 can be seen as -29 + 34 (which is 2×17) to give 5, so -29 is congruent to 5 modulo 17.
• Option IV: -3 can be seen as -3 + 20 (which is 1×17 + 3) to give 17, so -3 is not congruent to 5 modulo 17, as 17 modulo 17 gives 0, not 5.

Based on these calculations, the correct answer is (c) III, since only -29 is congruent to 5 modulo 17.