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What is the digit on the units place in the expanded value of 97^275 –

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Re: What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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New post 03 Jul 2019, 04:41
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The units digit of (97)^275 is 3
The units digit of (32)^44 is 6
Hence, the units digit of (97)^275 - (32)^44 is 7.

The answer is D.
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Re: What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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New post 03 Jul 2019, 06:34
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We are looking at the unit digits so we just need to consider the unit digit of each number.
So we need to look at
7^275 - 2^44
Now, unit digits of
7^1 = 7
7^2 = 9
7^3 = 3
7^4 = 1
7^5 = 7
So, we see after every 4th power the unit digit starts repeating.
We essentially need
7^3 as 275 = 4*68 + 3
Similarly
44 is a multiple of 4 so we will look at unit digit of 2^4
The problem is reduced to 3 - 6.
As 3 is a small number than 6 and is present at the units place, it will take a carry over from the tens place.
So, it becomes 13 - 6 = 7
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What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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New post Updated on: 03 Jul 2019, 08:34
97^275–32^44 units digits is will be same as units digits of xxxx7 ^ 275 - xxx2^44

Units digit of 7^275 = 7 and 2^44 = 6

Hence answer is 7-6 = 1.
Edit : Ah shoot my bad. Thanks for correcting me bebs. 7^275 units digit is 3. hence it is 3-6 , borrowing we get 13-6 = 7.
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Originally posted by laddaboy on 03 Jul 2019, 08:02.
Last edited by laddaboy on 03 Jul 2019, 08:34, edited 1 time in total.
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What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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New post 03 Jul 2019, 08:23
laddaboy Check your solution again. The unit digit for \(97^{275}\) is 3, not 7



laddaboy wrote:
97^275–32^44 units digits is will be same as units digits of xxxx7 ^ 275 - xxx2^44

Units digit of 7^275 = 7 and 2^44 = 6

Hence answer is 7-6 = 1.
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What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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New post 04 Jul 2019, 01:48
For finding the unit digit of any number, cyclic digit of that number needs to be known.
here, 7 has a cyclic rotation like 7, 9, 3, 1
2 has a cyclic rotation like 2, 4, 8, 6
so, 97^275 must be ended in 3 and 32^44 must be ended in 6
hence, the unit digit will be 7.
so, the correct answer choice is (D)
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Re: What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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New post 05 Jul 2019, 02:37
IMO D

7 is the unit digit of the 97 and have the cyclicity of 4 and then 275/4 remainder =3 that means the unit digit=3,

for the second number the end digit =6 (done similarly as above)

then the unit digit=7.
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What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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New post 05 Jul 2019, 12:57
Bunuel wrote:
What is the digit on the units place in the expanded value of \(97^{275} – 32^{44}\)?
A. 1
B. 3
C. 5
D. 7
E. 9
 

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This is the same as finding the units digit of 7ˆ275 - units digit of 2ˆ44;
Number 7 has a cycle of 4 {7,9,3,1}, and Number 2, also, has a cycle of 4 {2,4,8,6};
The units digit of 7ˆ275 will be the cycle number of the set equal to the remainder of 275/4, which is remainder 3, so its cycle num: 3;
The units digit of 2ˆ44 will be the cycle number of the set equal to the remainder of 44/4, which is remainder 0, so its the last cycle num: 6;
So, the units digit of 7ˆ275 - units digit of 2ˆ44 is equal to: 3 - 6 = -3, when we get a negative value, we subtract from 10, thus 10-3=7.

Answer (D).

PS: the cyclicality of a numbers units digit is given by that units digit raised to a power, for instance
[1] has a cycle of 1, because 1ˆ1, 1ˆ2, 1ˆ3… will always return 1;
[2] has a cycle of 4, because after 2 to an exponent divisible by 4 will always repeat the same units digit:
2ˆ1=2 has a units of 2,
2ˆ2=4 has a units of 4,
2ˆ3=8 has a units of 8,
2ˆ4=16 has a units of 6,
2ˆ5=32 has a units of 2 (here it begins to repeat the cycle).
2ˆ6=64 has a units of 4 etc…
[3] has a cycle of 4 {3,9,7,1}
[4] has a cycle of 2 {4,6}
[5] has a cycle of 1 {5}
[6] has a cycle of 1 {6}
[7] has a cycle of 4 {7,9,3,1}
[8] has a cycle of 4 {8,4,2,6}
[9] has a cycle of 2 {9,1}
the end.
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What is the digit on the units place in the expanded value of 97^275 –   [#permalink] 05 Jul 2019, 12:57

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