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Difficulty: Very hardUnit Digit and Cyclicity

What is the unit digit of the composite exponential expression E=332520×553213−448631E = 332^{520} \times 553^{213} - 448^{631}?

  1. A
    1
  2. B
    0
  3. 6Answer
  4. D
    4

Answer

The unit digit of the given expression is 6.
Evaluating each component using cyclicity of unit digits: 332520332^{520} has base unit digit 2 and exponent divisible by 4 (520 mod 4=0520 \bmod 4 = 0), so its unit digit is 24  ⟹  62^4 \implies 6. 553213553^{213} has base unit digit 3 with 213 mod 4=1213 \bmod 4 = 1, giving unit digit 31=33^1 = 3. 448631448^{631} has base unit digit 8 with 631 mod 4=3631 \bmod 4 = 3, giving unit digit 83  ⟹  28^3 \implies 2. Combining these gives (6×3)−2=18−2=16(6 \times 3) - 2 = 18 - 2 = 16, which yields a final unit digit of 6.

Step-by-Step Solution

1
Determine the unit digit of 332520332^{520} using cyclicity of 2
Unit digit is 6
The unit digit of base 332 is 2. The cyclicity of 2 is 4 (2, 4, 8, 6). Dividing the exponent 520 by 4 gives a remainder of 0. When remainder is 0, we take the 4th power: 24=162^4 = 16, so the unit digit is 6.
2
Determine the unit digit of 553213553^{213} using cyclicity of 3
Unit digit is 3
The unit digit of base 553 is 3. The cyclicity of 3 is 4 (3, 9, 7, 1). Dividing the exponent 213 by 4 gives a remainder of 1 (213=4×53+1213 = 4 \times 53 + 1). Thus, the unit digit is 31=33^1 = 3.
3
Determine the unit digit of 448631448^{631} using cyclicity of 8
Unit digit is 2
The unit digit of base 448 is 8. The cyclicity of 8 is 4 (8, 4, 2, 6). Dividing the exponent 631 by 4 gives a remainder of 3 (631=4×157+3631 = 4 \times 157 + 3). Thus, the unit digit is 83=5128^3 = 512, which ends in 2.
4
Combine the unit digits according to the expression E=(332520×553213)−448631E = (332^{520} \times 553^{213}) - 448^{631}
Unit digit is 6
First multiply the unit digits of the first two terms: 6×3=18  ⟹  86 \times 3 = 18 \implies 8. Then subtract the unit digit of the third term: 8−2=68 - 2 = 6.

Key Concept

Unit Digit Cyclicity Rules
Estimated Time:1m 30s
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Topics

Unit Digit and Cyclicity

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