Q74·CSAT · Prelims 2023
Remainder of 2^192 divided by 6
NumericalRemainders & CyclicityNumber theory● Medium
Question
What is the remainder if 2^192 is divided by 6?
Options
a
0
b
1
c
2
dAnswer
4
Explanation
Analyze the behavior of consecutive small integer powers of 2 modulo 6: 2^1 \equiv 2 ± od 6 2² = 4 \equiv 4 ± od 6 2³ = 8 \equiv 2 ± od 6 2^4 = 16 \equiv 4 ± od 6
The remainder cycle displays an oscillating pattern between 2 and 4 for all positive integer powers.
For any odd exponent (n = 1, 3, 5, \dots), the remainder is strictly 2.
For any even exponent (n = 2, 4, 6, \dots), the remainder is strictly 4.
Since the target exponent 192 is an even integer, the remainder must be exactly 4.
For cyclic remainder sequences where the base shares common prime factors with the modulus, identify the repeating loop period and map the parity of the exponent directly to its slot in the cycle.
Answer: (d).
Question details
Year
2023
Paper
CSAT
Question
Q74
Section
Numerical Ability
Sub-topic
Remainders & Cyclicity
Type
Number theory
Difficulty
Medium
Source hint
Number theory
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