MrVarghese wrote:
What is the unit digit of (88^77-77^88)?
A) 7
B) 3
C) 4
D) 1
D) 0
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 powerso 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
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