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04 Jan 2026, 22:00
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B
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Difficulty:
25%
(medium)
Question Stats:
75% (01:25) correct
25%
(01:23)
wrong
based on 137
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What is the digit at the units place in the total integer value of 19^31 – 18^16 ?
A. 1
B. 3
C. 5
D. 7
E. 9
A. 1
B. 3
C. 5
D. 7
E. 9
05 Jan 2026, 06:33
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To find the units digit of 19^31 - 18^16, we need to analyze the cyclicity of the units digits for the bases. Remember, we only care about the last digit of the base numbers.
Step 1: Analyze 19^31 We look at the base 9. The powers of 9 follow a 2-step cycle: 9^1 = 9 9^2 = 81 (ends in 1) 9^3 = ...9
Rule:
Step 2: Analyze 18^16 We look at the base 8. The powers of 8 follow a 4-step cycle: 8^1 = 8 8^2 = ...4 8^3 = ...2 8^4 = ...6 8^5 = ...8 (Cycle repeats)
Rule: Divide the exponent by 4 to find the position in the cycle. 16 / 4 gives a remainder of 0.
A remainder of 0 means we take the last digit in the cycle (the 4th value). Therefore, the units digit of 18^16 is 6.
Step 3: Final Calculation Subtract the units digits found in the steps above: 9 - 6 = 3
Correct Answer: B
Step 1: Analyze 19^31 We look at the base 9. The powers of 9 follow a 2-step cycle: 9^1 = 9 9^2 = 81 (ends in 1) 9^3 = ...9
Rule:
- Odd Exponent means the units digit is 9
- Even Exponent means the units digit is 1
Step 2: Analyze 18^16 We look at the base 8. The powers of 8 follow a 4-step cycle: 8^1 = 8 8^2 = ...4 8^3 = ...2 8^4 = ...6 8^5 = ...8 (Cycle repeats)
Rule: Divide the exponent by 4 to find the position in the cycle. 16 / 4 gives a remainder of 0.
A remainder of 0 means we take the last digit in the cycle (the 4th value). Therefore, the units digit of 18^16 is 6.
Step 3: Final Calculation Subtract the units digits found in the steps above: 9 - 6 = 3
Correct Answer: B
05 Jan 2026, 22:41
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What is the digit at the units place in the total integer value of 19^31 – 18^16 ?
Understand that in order to determine the units place, we do not need to calculate the full values of the exponents; we can get away with figuring out only the units digit.
Let us consider each of the exponents separately -
19 ^ 31 - here instead of looking at 19 as a whole we can only take 9 and consider the units digit of 9.
We have to check the pattern for the units digit occurence for powers of 9 -
9 ^ 1 = unit digit is 9
9 ^ 2 = unit digit is 1
9 ^ 3 = unit digit is 9
9 ^ 4 = unit digit is 1
so there is a repetitive pattern of either 9 or 1. As 31 is an odd number the unit digit will be = 9.
Now let us consider 18 ^ 16.
we will follow the same as above.
8 ^ 1 = unit digit is 8
8 ^ 2 = unit digit is 4
8 ^ 3 = unit digit is 2
8 ^ 4 = unit digit is 6
8 ^ 5 = unit digit is 8
8 ^ 6 = unit digit is 4
8 ^ 7 = unit digit is 2
8 ^ 8 = unit digit is 6
here the repetitive pattern is of 8, 4, 2 or 6. As 16 is a multiple of 4 the units digit will be 6.
Final answer is ----> 9 - 6 = 3.
Understand that in order to determine the units place, we do not need to calculate the full values of the exponents; we can get away with figuring out only the units digit.
Let us consider each of the exponents separately -
19 ^ 31 - here instead of looking at 19 as a whole we can only take 9 and consider the units digit of 9.
We have to check the pattern for the units digit occurence for powers of 9 -
9 ^ 1 = unit digit is 9
9 ^ 2 = unit digit is 1
9 ^ 3 = unit digit is 9
9 ^ 4 = unit digit is 1
so there is a repetitive pattern of either 9 or 1. As 31 is an odd number the unit digit will be = 9.
Now let us consider 18 ^ 16.
we will follow the same as above.
8 ^ 1 = unit digit is 8
8 ^ 2 = unit digit is 4
8 ^ 3 = unit digit is 2
8 ^ 4 = unit digit is 6
8 ^ 5 = unit digit is 8
8 ^ 6 = unit digit is 4
8 ^ 7 = unit digit is 2
8 ^ 8 = unit digit is 6
here the repetitive pattern is of 8, 4, 2 or 6. As 16 is a multiple of 4 the units digit will be 6.
Final answer is ----> 9 - 6 = 3.
