Which of the following statements are true? I. If a ≡ b (mod n), then a^2 ≡ b^2 (mod n). II. If ac ≡ bc (mod n), then always a ≡ b (mod n). III. If a ≡ b (mod n), then b ≡ a (mod n). IV. If a ≡ b and b ≡ c (mod n), then a ≡ c (mod n).

Question 57

MediumOriginal question

Which of the following statements are true?
I. If $a \equiv b \pmod{n}$ , then $a^{2} \equiv b^{2} \pmod{n}$ .
II. If $ac \equiv bc \pmod{n}$ , then always $a \equiv b \pmod{n}$ .
III. If $a \equiv b \pmod{n}$ , then $b \equiv a \pmod{n}$ .
IV. If $a \equiv b$ and $b \equiv c \pmod{n}$ , then $a \equiv c \pmod{n}$ .

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

Show the answer and explanation

CorrectNot quiteAnswer B

Correct answer: B

I: congruences can be multiplied. III and IV: congruence is symmetric and transitive.
II is false: $2 \times 3 \equiv 4 \times 3 \pmod{6}$ (both are $\equiv 0$ ), but $2 \not\equiv 4 \pmod{6}$ .

Worth noting

Cancelling $c$ is safe only when $c$ and $n$ have no common factor.

Concept: Properties of congruence
Typical timeTypical: 120 s
Your timeYours: —

The rule this testsCongruence modulo n

Practise this chapter →

Ten questions from Modulo Arithmetic & Congruence, at your own pace.

Explanation