Difficulty: Easy
Correct Answer: 0
Explanation:
Introduction / Context:
This number systems question focuses on finding the units digit of very large powers, which is a common task in aptitude exams. Instead of calculating the full value of 194^102 and 294^103, which is impossible by hand, you only need to track the pattern of the last digit in powers of a given base. Recognising these cyclic patterns is a powerful shortcut for handling large exponents.
Given Data / Assumptions:
Concept / Approach:
The key idea is that the units digit of a power depends only on the units digit of its base. Both 194 and 294 end with the digit 4, so we analyse the pattern of the units digit for 4^n. Once we know the units digit for 4^102 and 4^103, we add them and extract the final units digit of the sum.
Step-by-Step Solution:
Verification / Alternative check:
As another confirmation, observe that only the last digits 4 and 4 matter, and the exponents 102 and 103 fall into the same cycle pattern modulo 2. You can check 4^2 and 4^3 directly: 4^2 ends with 6 and 4^3 ends with 4, whose sum ends with 0. This small example mirrors the larger exponents and validates the reasoning.
Why Other Options Are Wrong:
The digits 6, 8, 2, and 4 would result only if the pattern of powers of 4 were misunderstood. For instance, assuming a cycle length of 4 or misidentifying even and odd exponents could lead to these wrong units digits. Correctly tracking the two term pattern 4, 6, 4, 6 eliminates these possibilities.
Common Pitfalls:
Common mistakes include trying to calculate large powers directly, which is impractical, or assuming a longer cycle than actually exists for the digit 4. Some learners also forget to simplify the problem by noting that 194 and 294 share the same units digit. Focusing on patterns of last digits is the best way to solve such problems efficiently.
Final Answer:
The digit in the units place of 194^102 + 294^103 is 0.
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