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CAT 2024 Slot 2 — QA Question 13

Remainder of a to the power n, divided by b.Easy

When 3333 is divided by 11, the remainder is

Answer & solution

Correct answer: 5

  • 5

  • B

    6

  • C

    1

  • D

    10

Solution

Easy

Find the order of 33 modulo 1111: a small power of 33 that is ≡1\equiv 1. Then reduce the exponent 333333 modulo that order.

1

Find a cycle. Compute powers of 33 mod 1111.

31≡3, 32≡9, 33≡27≡5, 34≡15≡4, 35≡12≡1(mod11)\begin{aligned} &3^1\equiv3,\ 3^2\equiv9,\ 3^3\equiv27\equiv5,\ 3^4\equiv15\equiv4,\ 3^5\equiv12\equiv1\pmod{11} \end{aligned}
2

Reduce the exponent. Since 35≡13^5\equiv1, work with 333 mod 5333\bmod 5.

333=5×66+3 ⇒ 333≡3(mod5)⇒ 3333≡33(mod11)\begin{aligned} &333=5\times 66+3\ \Rightarrow\ 333\equiv 3\pmod 5\\ &\Rightarrow\ 3^{333}\equiv 3^{3}\pmod{11} \end{aligned}
3

Read off the remainder.

33=27≡5(mod11)\begin{aligned} &3^{3}=27\equiv 5\pmod{11} \end{aligned}
Remainder=5\text{Remainder}=5

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CAT 2024 Slot 2 QA Q13 — Remainder of a to the power n, divided by b. solution | TheCATExam