Assertion (A): 2^10 ≡ 1 (mod 11). Reason (R): 2^5 = 32 ≡ -1 (mod 11).

Modulo Arithmetic & Congruence · Question 59 of 97

Question 59

MediumOriginal question

Assertion (A): $2^{10} \equiv 1 \pmod{11}$ .
Reason (R): $2^{5} = 32 \equiv -1 \pmod{11}$ .

Both A and R are true, and R is the correct explanation of A.Correct answerYour answerBoth A and R are true, but R is not the correct explanation of A.Correct answerYour answerA is true, but R is false.Correct answerYour answerA is false, but R is true.Correct answerYour answer

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CorrectNot quiteAnswer A

Correct answer: A

R: $32 = 33 - 1$ , and $11 \mid 33$ . True.
Squaring R: $2^{10} = (2^{5})^{2} \equiv (-1)^{2} = 1$ , so A is true and follows from R.

Concept: Congruence to a negative number
Typical timeTypical: 90 s
Your timeYours: —

The rule this testsCongruence modulo n

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Assertion (A): 210≡1(mod11)2^{10} \equiv 1 \pmod{11}210≡1(mod11).Reason (R): 25=32≡−1(mod11)2^{5} = 32 \equiv -1 \pmod{11}25=32≡−1(mod11).

Explanation

Assertion (A): $2^{10} \equiv 1 \pmod{11}$ .
Reason (R): $2^{5} = 32 \equiv -1 \pmod{11}$ .