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17 Jan 2012, 17:52
Kudos
Bookmarks
Hey friends i have minor doubt can you please clarify this?
What is the units digit of 4^100?
->To my knowledge we generally go with the the number to the power of units digit
Eg:3^84----->since the it follows a sequence such as 3,9,27,81,243......so the units digits follows a sequence 3,9,7,1,3..........so the units digit of 3^84 should be 1....
Now look at the question i.e units digit of 4^100
since units digit sequence of 4 varies in the form 4,6,4........but here the question says 4^0(as the units digit is 0)
so my doubt that are we supposed to consider 4^0 if this is the case then units digit is 1....But Mnhattan Gmat says the units digit of 4^100 is 4...How come this is true?
Its in page 58 Manhattan 12th problem (03 - The Equations, Inequalities, and VICs Guide 4th edition)
Can anyone please explain??
What is the units digit of 4^100?
->To my knowledge we generally go with the the number to the power of units digit
Eg:3^84----->since the it follows a sequence such as 3,9,27,81,243......so the units digits follows a sequence 3,9,7,1,3..........so the units digit of 3^84 should be 1....
Now look at the question i.e units digit of 4^100
since units digit sequence of 4 varies in the form 4,6,4........but here the question says 4^0(as the units digit is 0)
so my doubt that are we supposed to consider 4^0 if this is the case then units digit is 1....But Mnhattan Gmat says the units digit of 4^100 is 4...How come this is true?
Its in page 58 Manhattan 12th problem (03 - The Equations, Inequalities, and VICs Guide 4th edition)
Can anyone please explain??
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17 Jan 2012, 18:33
Kudos
Bookmarks
Hi, there. I'm happy to help with this. 
First of all, the patterns of what the unit digits of powers are only hold for positive integer powers, powers n > 0. Any integer to the power of 0 is 1, and at n = 0, that pattern trumps whatever individual pattern the individual base has.
As for powers of 4, you are perfectly correct: the unit digits of the powers are 4, 6, 4, 6, 4, 6 ---> 4^(odd power) has a unit's digit of 4, and 4^(even power) has a unit's digit of 6. Thus, 100 is even, so 4^100 would have a units digit of 6.
I am holding the Manhattan GMAT book to which you referred, and I believe the question you cited is p. 66, #12: "If Sn = 4^n + 5^(n+1) + 3, what is the units digit of S100?" That question ---- not simply the question about the units digit of 4^100 by itself ---- has an answer of 4. The reason is: the 4^100 part would have units digit of 6, any power of 5 has a units digit of 5, and 6 + 5 + 3 = 14, something with a units digit of 4.
Does that all make sense? I hope that clears up your confusion. Please let me know if you have any further questions.
Mike
First of all, the patterns of what the unit digits of powers are only hold for positive integer powers, powers n > 0. Any integer to the power of 0 is 1, and at n = 0, that pattern trumps whatever individual pattern the individual base has.
As for powers of 4, you are perfectly correct: the unit digits of the powers are 4, 6, 4, 6, 4, 6 ---> 4^(odd power) has a unit's digit of 4, and 4^(even power) has a unit's digit of 6. Thus, 100 is even, so 4^100 would have a units digit of 6.
I am holding the Manhattan GMAT book to which you referred, and I believe the question you cited is p. 66, #12: "If Sn = 4^n + 5^(n+1) + 3, what is the units digit of S100?" That question ---- not simply the question about the units digit of 4^100 by itself ---- has an answer of 4. The reason is: the 4^100 part would have units digit of 6, any power of 5 has a units digit of 5, and 6 + 5 + 3 = 14, something with a units digit of 4.
Does that all make sense? I hope that clears up your confusion. Please let me know if you have any further questions.
Mike
18 Jan 2012, 04:31
Kudos
Bookmarks
kotela
The unit's digit of the base is the one we are focusing on. Not the unit's digit of the power.
Say, you have \(264^{100}\) and you need its unit's digit. How will you approach it?
The unit's digit will be determined by the unit's digit of the base i.e. 4.
Next, the power is 100.
The cyclicity of 4 is 2 i.e. the unit's digit of 4 varies as: 4, 6, 4, 6, 4, 6...
If 4 has an odd power (say \(4^1 or 4^3 or 4^{257}\)), the unit's digit will be 4
If 4 has an even power ((say \(4^2 or 4^8 or 4^{256}\)), the unit's digit will be 6.
100 is even so \(4^{100} = 6\)
