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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 02 Jul 2019, 10:51
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7^275 will result with an end unit of 3
2^44 will result with an end unit of 6

3-6 = 7, D
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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, 01:19
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Find the units digit pattern as follows:

..7^1 = units digit 7
..7^2 = units digit 9
..7^3 = units digit 3
..7^4 = units digit 1
..7^5 = units digit 7
..7^6 = units digit 9, etc

Thus, 97^275 will have units digit of 3

32^1 = units digit 2
32^2 = units digit 4
32^3 = units digit 8
32^4 = units digit 6
32^5 = units digit 2
32^6 = units digit 4, dst.

Thus, 32^44 will have a units digit of

97^275-32^44 =...3 -...6 =...7

Answer is (D)

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Originally posted by chondroht on 02 Jul 2019, 11:30.
Last edited by chondroht on 03 Jul 2019, 01:19, edited 1 time in total.
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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 02 Jul 2019, 11:40
Cyclicity(last digit has repeats after every 4) of 7 is 4, i.e
\(7^1 = 7\)
\(7^2 = 49\)
\(7^3 = 343\)
\(7^4 = 2401\)

275=68*4+3 means its the third in the cycle
Unit's digit of \(97^275\) is 3

Cyclicity of the digit 2 is 4
\(2^1 = 2\)
\(2^2 = 4\)
\(2^3 = 8\)
\(2^4 = 16\)

44=11*4 means its the 4th in the cycle
Units Digit of 32^44 is 6

3-6=3

Answer B
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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 02 Jul 2019, 11:42
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97^275–32^44

97^275 - Unit's digit 7 has a cyclicity of 4. => unit's digit of 97^275 is equivalent to the unit's digit of 7^3
=> Unit's digit of 97^275 = 3

32^44 - Unit's digit 2 has a cyclicity of 4. => unit's digit of 32^44 is equivalent to the unit's digit of 2^4
=> Unit's digit of 32^44 = 6

3-6 = -3 which is equivalent to 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 02 Jul 2019, 12:21
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We can begin with routine work: Calculating cyclicity
\(7^1=7\)
\(7^2=49\)
\(7^3=..3\)
\(7^4=..1\)

275/4 leaves remainder 3. The units digit of \(97^{275}\) will be 3

\(2^1=2\)
\(2^2=4\)
\(2^3=8\)
\(2^4=16\)

44/4 leaves no remainder. The units digit of \(32^{44}\) will be 6

here if you think that only 3-6 is left to calculate BE WARE
ALWAYS check which number is greater in such cases
In our case \(97^{275}\)>\(32^{44}\) and we can safely deduce that ..3-..6=..7
But if the first number is less then we can have different units digit: 13-26=-13

IMO
Ans: 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 02 Jul 2019, 12:31
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97^275: the unit digit will be 3.

As any multiple of 7 will have unit digits as 7, 9, 3, 1. This cycle repeats for every 4th exponent.

32^44: the unit digit will be 6.

As any multiple of 2 will have unit digits as 2, 4, 8, 6. This cycle repeats for every 4th exponent.

3 - 6 = unit digit will be 7.
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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 02 Jul 2019, 12:50
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Unit digit for 97^275 is 3
unit digit for 32^44 is 6
3-6 = 7
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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 02 Jul 2019, 12:51
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Answer: D

7 has a cyclicity of 4, so last digit of 97^275 --> is 3, since 275 can be written as 68*4+3
2 has a cyclicity of 4, so 32^44, last digit --> 6
3-6 = -3 as units digit cannot be negative, subtract 10, which is carried over. So, 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 02 Jul 2019, 14:15
7^1 = 7
7^2 = 49
7^3 = ends with 3
7^4 ends with 1
7^5 again ends with 7

2^2 = 4
2^3 = 8
2^4 = ends with 6
2^5 again ends with 2

So 97^275 will have some value as 7^5 which has 7 in units place
and 32^44 will have some value 2^(11*4) which has 6 in the units place

Hence, 7-6 = 1.
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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 02 Jul 2019, 14:20
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Start printing pattern.
For first term its 7,9,3,1
Second term tis 2,4,8,6

if first term is multiplied 275 times its goes 68 times with units digit and three additional times giving 3 as units digit.
Similar second term is 6.

13-6 =7
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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 02 Jul 2019, 14:30
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97^275 - 32^44

Following 7's cyclicity of 4 -

7^1 = 7 = ending in 7
7^2 = 49 = ending in 9
7^3 = 343 = ending in 3
7^4 = 2401 = ending in 1

So, 97^275 will be like

7^275 since we are dealing only with digit in units place currently.

Hence,

7^275 is of the type 7^3 and hence will end in a 3

Similarly, following 2's cyclicity of 4 -

2^1 = 2 = ending in 2
2^2 = 4 = ending in 4
2^3 = 8 = ending in 8
2^4 = 16 = ending in 6

So, 32^44 will be like

2^44 since we are dealing only with digit in units place currently.

Hence,

2^44 is of the type 2^4 and hence will end in a 6

Now we have 2 numbers each ending in 3 and 6 respectively.

A difference of the 2 numbers will leave a 7 in the units digit of the resulting number.
Example: 153 - 46 = 107

Hence answer is 7 and option D
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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: 04 Jul 2019, 01:55
Made a silly mistake in my working out.

7-9-3-1
7^275 = 272 (highest multiple of 4 that would repeat the cycle) + 3 = 7-9-3-1

2^(some multi of 4) = cyclicity of 4 = 2-4-8-16

xxxx3
xxx6

We need to treat the 3 as part of a larger number (Which it is) so we end up with 13 - 6 (in performing the subtraction) to get 7
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Originally posted by dcummins on 02 Jul 2019, 15:13.
Last edited by dcummins on 04 Jul 2019, 01:55, edited 1 time in total.
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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 02 Jul 2019, 15:38
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Cyclicity of digits.

Units digits :-
7^1 = 7
7^2 = 9 (49)
7^3 = 3 (343)
7^4 = 1 (2401)

As we can see, 7^4 has 1 at its units digit. Thus 7^5 will have a 7 at its units digit and the whole cycle will be repeated.

Now, 97 has 7 at its units place and the 7 will dictate the units digit of all its exponents, i.e. 97^1 has 7 at its units digit, 97^2 will have 9(9409) at its units digit and so on.
Thus 97^275 = [97^(272)]*[97^3]
272 = 4*68
Units digit of 97^272 = 1
Units digit of 97^3 = 3

Thus units digit of 97^275 = 3


Similarly,
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 6 (16)
2^5 = 2 (32)

So, units digit of 32^44 will be the same as the units digit of 2^44
Which is the same as the units digit of 2^4 = 6.

So the units digit of the given equation = 3 - 6 = 7.
Hence (D)


P.S. since 97^275 is clearly greater than 32^44, the resulting number when 32^44 is subtracted from 97^275 will be positive.
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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 02 Jul 2019, 17:38
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What is the digit on the units place-

D, The unit digit o the expression determines the unit digit of the answer-
Cyclicity of 2 and 7 is 4, so the expression is basically reduced to:
7^3 - 2^4 = 3-6 = 7

Note: 275= 4K + 3
& 44= 4K'
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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 02 Jul 2019, 18:06
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This one is about cyclicity , so you have to consider cyclicity of the unit digits (7 and 2) in order to find the resultant unit digit:

cyclicity units of 7:

7^1=7
7^2=9
7^3=3
7^4=1

cyclicity units of 2:

2^1=2
2^2=4
2^3=8
2^4=6

So, for 97^275:

275=4K+3, so the unit number will have 3 as units

So, for 32^44:

44=4K, so the unit number will have 6 as units

Then, since 97^275 > 32^44, and the units of each are 3 and 6 respectively, the resultant unit digit must be 7.

D is the answer.
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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 02 Jul 2019, 18:31
7 repeats its unit digits, every 5 numbers-
Units digits-
7^1=7,7^2=9, 7^3=3, 7^4=1, 7^5=7

Thus, 7^275 have a units digit as 7

Similarly, for 2, 2 repeats its unit digits every 5th power (2,4,8,6,2)
Thus, 2^44= 2^(40+4), thus unit digit as 2^4= 6

7-6=1
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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 02 Jul 2019, 18:48
First of all, look at 97^275, we don't really need to get the full number, all we need to know is the number in the unit place. So we already know the first number in the unite place is 7. Then 97X97, we multiple the number in the unit place 7X7 first, it's equal to 49, so for the second number, the unit place would be 9, we don't need to get the full answer. Then for the next number, no matter what answer we get previously, when it times to 97, the number in unit place would be 9 (the unit digit from last number) X7=63, so number in unit place would be 3. Then times 97 again, 3X7=21, the unit digit would be 1. Then we multiple 97 again, finally we see a train in here, the unit digit will cycles in 4 numbers which are 7,9,3 and 1.

Second, look at 32^44, used the same way and we will see that the unit digit will also cycles in 4 numbers: 2,4,8 and 6

Finally, we used 7-2, 9-4, 8-3 and 6-1 all the answer are the same, which is number 5

So choose answer C
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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 02 Jul 2019, 18:53
First, let examize the last unit digit of it:

97^275

We have the reoccurring pattern of the last digit of 7^x:
7^1 ends with 7
7^2 ends with 9
7^3 ends with 3
7^4 ends with 1
7^5 ends with 7
...
So the pattern 7,9,3,1,7,9,3,1... will repeat every four 7^x
Plus, we have 275/4=68 with the remainder of 3. So 97^275 ends with 3

Similarly, 32^44 has the repeating pattern of last digit of 2,4,8,6,2...
We have 44/4=11, no remainder, so 32^44 ends with 2.

The we have 97^275 - 32^44 ends with 3-2=1

I do with A

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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 02 Jul 2019, 19:00
This is a simple problem on cyclic nature of last digit of subsequent powers of a number.
Cyclicity of last digit for numbers ending with 7 - 7, 9, 3, 1,7..... (cyclicity=4) and for numbers ending with 2 - 2,4,8,6,2...(cyclicity=4)
here, dividing power by cyclicity will leave a remainder, which will allow us to choose the last digit for that power
275/4 leaves 3 as a remainder, so the last digit for this term will be 3
similarly the right term has 0 remainder and this corresponds to last digit 6
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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 02 Jul 2019, 19:32
Answer is C because we use the cyclicity of unit digit.

First number ends with 7 whose cyclicity is 4 => end at 3
Second number ends with 2 whose cyclicity is 4 => end at 8
=> 3-8 = 5
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Re: What is the digit on the units place in the expanded value of 97^275 –   [#permalink] 02 Jul 2019, 19:32

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