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Updated on: 28 Mar 2026, 13:50
Originally posted by MrVarghese on 30 Aug 2019, 06:17.
Last edited by Bunuel on 28 Mar 2026, 13:50, edited 3 times in total.
Last edited by Bunuel on 28 Mar 2026, 13:50, edited 3 times in total.
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
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23% (01:41) correct
77%
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What is the unit digit of \(88^{77} - 77^{88}\)?
A. 7
B. 3
C. 4
D. 1
D. 0
A. 7
B. 3
C. 4
D. 1
D. 0
Updated on: 28 Mar 2026, 13:48
Originally posted by freedom128 on 30 Aug 2019, 14:09.
Last edited by Bunuel on 28 Mar 2026, 13:48, edited 12 times in total.
Last edited by Bunuel on 28 Mar 2026, 13:48, edited 12 times in total.
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Q. Units digit of 88^77−77^88
1) 88^77 = 88^{4∗19+1}. Thus, 88^77 has same units digit as 88^1 and 8^1, that is 8.
2) 7^88 = 77^{4∗22}. Thus, 77^88 has same units digit as 77^4 and 7^4, that is 1.
Therefore, units digit of 88^77−77^88 could be either:
(1) 8-1=7 (if 88^77 > 77^88), or
(2) 8-11=-3 (if 88^77 < 77^88).
Obviously, Case 2) makes much more sense to me. So the unit digit of 88^77−77^88 must be 8-11=-3.
Answer is (B)
1) 88^77 = 88^{4∗19+1}. Thus, 88^77 has same units digit as 88^1 and 8^1, that is 8.
2) 7^88 = 77^{4∗22}. Thus, 77^88 has same units digit as 77^4 and 7^4, that is 1.
Therefore, units digit of 88^77−77^88 could be either:
(1) 8-1=7 (if 88^77 > 77^88), or
(2) 8-11=-3 (if 88^77 < 77^88).
Obviously, Case 2) makes much more sense to me. So the unit digit of 88^77−77^88 must be 8-11=-3.
Answer is (B)
General Discussion
30 Aug 2019, 09:32
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MrVarghese
The units digit have a cyclicity ..
When we talk of units digit, we should know the following rules of cyclicity..
every digit has cyclicity when it comes to last/unit's digit...each digit surely repeats the units digit after every 4th power..
i) 1, 5 and 6 and 0 repeat with each power..
1^1 or 1^2 will always give 1 and similarly for 5,6,0
ii) 4,9 repeat after increase of two powers..
4^1 or 4^3 or 4^5 will all leave 4 as last digit and 4^2 or 4^4 will leave 6 as last digits
so cyclicity is 4,6,4,6..
for 9 it is 9,1,9,1,9...
iii) remaining 2,3,7,8 repeat after every 4th power
so 2^1 , 2^(1+4), 2^(1+4+4) will leave 2 in each case ..cyclicity is 2,4,8,6,2,4,8,6..
for 3 it is 3,9,7,1,3,9,..
for 7 it is 7,9,3,1,7,9,3,1...
for 8 it is 8,4,2,6,8,4,2,6..
So now we can look at the question..
Units digit of \(88^{77}-77^{88}\)..
1) \(88^{77}\) will have same units digit as \(8^{77}=8^{4*19+1}\), which will have same units digit as \(8^{1}\)
2) \(77^{88}\) will have same units digit as \(7^{88}=7^{4*22}\), which will have same units digit as \(7^{4}\), that is 1
so Units digit of \(88^{77}-77^{88}\) = 8-1=7, if \(88^{77}>77^{88}\) or 8-11=-3, if \(88^{77}<77^{88}\)
Now, is there any simple method to check which one is greater? MAY not be, and therefore such values will not be tested in gmat.
But here it is case 2, where \(88^{77}<77^{88}\), so answer will -abcde3
B
