Which of the following statements are true? I. a mod n is always one of 0, 1, , n - 1. II. (-1) mod n = n - 1. III. (-a) mod n = a mod n for every integer a. IV. a mod n = 0 exactly when n divides a.

Modulo Arithmetic & Congruence · Question 11 of 97

Question 11

MediumOriginal question

Which of the following statements are true?
I. $a \bmod n$ is always one of $0, 1, \dots, n - 1$ .
II. $(-1) \bmod n = n - 1$ .
III. $(-a) \bmod n = a \bmod n$ for every integer $a$ .
IV. $a \bmod n = 0$ exactly when $n$ divides $a$ .

All four: I, II, III and IVCorrect answerYour answerI and IV onlyCorrect answerYour answerII, III and IV onlyCorrect answerYour answerI, II and IV onlyCorrect answerYour answer

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CorrectNot quiteAnswer D

Correct answer: D

I is the definition of a remainder, and IV says the remainder is 0. For II, $-1 = n \times (-1) + (n - 1)$ .
III is false: $3 \bmod 5 = 3$ but $(-3) \bmod 5 = 2$ .

Concept: Facts about the mod operator
Typical timeTypical: 120 s
Your timeYours: —

The rule this testsThe modulo operation

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