Arrange the following in ascending order: (A) Remainder of 4916 when divided…
2023
Arrange the following in ascending order:
(A) Remainder of 4916 when divided by 17
(B) Remainder of 2446 when divided by 9
(C) Remainder of 15517 when divided by 17
(D) Last digit of the number 745
Choose the correct answer from the options given below:
Answer: C. (A), (C), (B), (D) — ConceptFor modular powers, first reduce the base modulo the divisor, then reduce the exponent using a valid power cycle or Fermat’s little theorem. For a last…
- A.
(A), (B), (C), (D)
- B.
(A), (B), (D), (C)
- C.
(A), (C), (B), (D)
- D.
(D), (C), (B), (A)
Attempted by 35 students.
Show answer & explanation
Correct answer: C
Concept
For modular powers, first reduce the base modulo the divisor, then reduce the exponent using a valid power cycle or Fermat’s little theorem. For a last digit, work modulo 10 and use the repeating last-digit cycle.
After evaluating each expression independently, compare the four numerical results to arrange them from least to greatest.
Application
For (A), 49 ≡ −2 (mod 17). Since 17 is prime, Fermat’s little theorem gives 216 ≡ 1 (mod 17). Therefore, 4916 leaves remainder 1.
For (B), powers of 2 modulo 9 repeat every 6 powers. Since 446 = 6 × 74 + 2, we get 2446 ≡ 22 ≡ 4 (mod 9). Therefore, the remainder is 4.
For (C), 155 ≡ 2 (mod 17). Using 216 ≡ 1 (mod 17), we obtain 217 = 2 × 216 ≡ 2 (mod 17). Therefore, 15517 leaves remainder 2.
For (D), the last digits of powers of 7 cycle as 7, 9, 3, 1. Since 45 = 4 × 11 + 1, 745 has last digit 7.
Thus the values are (A) = 1, (B) = 4, (C) = 2, and (D) = 7. Sorting these gives 1 < 2 < 4 < 7, so the ascending order is (A), (C), (B), (D).
Cross-check
The reduced values 1, 2, 4, and 7 are already strictly increasing in the sequence (A), (C), (B), (D), confirming the ordering without any tie.
Result: (A), (C), (B), (D).