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What is the unit digit of the expression (234)45 × (567)34?
2
4
6
8
6
To find the unit digit of (234)45, consider the cycle of 4, which is 4, 6 (repeating every 2); since 45 is odd, the unit digit is 4. For (567)34, the cycle of 7 is 7, 9, 3, 1; since 34 mod 4 = 2, the unit digit is 9. Multiplying 4 and 9 gives 36, so the final unit digit is 6.
To find the unit digit of (234)45, consider the cycle of 4, which is 4, 6 (repeating every 2); since 45 is odd, the unit digit is 4. For (567)34, the cycle of 7 is 7, 9, 3, 1; since 34 mod 4 = 2, the unit digit is 9. Multiplying 4 and 9 gives 36, so the final unit digit is 6.
The concept of modular arithmetic used here underpins the RSA encryption algorithm, which is a vital component of India's Digital India initiative and secure e-governance infrastructure.
The unit digit of 4 raised to an odd power is always 4.
The unit digit of 4 raised to an even power is always 6.
The power cycle of 7 is 4 (7, 9, 3, 1).
Calculation: Unit digit of 234⁴⁵ = 4 (because 45 is odd).
Calculation: Unit digit of 567³⁴ = 9 (34 divided by 4 leaves remainder 2, 7² = 49).
Final result: 4 * 9 = 36, hence the unit digit is 6.
The official answer provided (Option A) is mathematically incorrect based on standard cyclic properties.
C is correct — The calculated unit digit of the expression (234)45 × (567)34 is 6.
Always verify cyclicity rules for digits 2, 3, 7, and 8, as they share a cycle of 4, which frequently appears in RRB and SSC quantitative aptitude sections.
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