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

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

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02 Jul 2019, 08:00
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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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Re: What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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02 Jul 2019, 14:05
4
Any number ending in 7 when raised to a power will have the following pattern 7,9,3,1 as the units digit

and Any number ending in 2 when raised to a power will have the following pattern 2,4,8,6 as the units digit

Now 97^275 means we divide 275 by 4 and compare it against the pattern. 275th power will have 3 as the units digit

and
Now 32^44 means we divide 44 by 4 and compare it against the pattern. 44th power will have 6 as the units digit

Thus we have 3 - 6 --->>> The trick is that you have to imagine the normal subtraction and get 1 as the carry over. thus it is actually 13 - 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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02 Jul 2019, 08:13
3
The cyclicity of 7 and 2 is 4.That is the last digit repeats after every four powers.

Say for 7
7
9
3
1
again
7
9
3
1
Similarly for 2 it is 2,4,8,6

Now
97^275 implies last digit is 7 ^ 3= 3 (275/4 leaves 3 )

32^44 implies last digit is 6 (44/2 leaves 0 )

Therefore last digit = 3-6 = 13-6= 7 (last digit will not be negative we will have to borrow one )
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What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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Updated on: 02 Jul 2019, 09:39
2
Cyclicity of 7 is 4 so $$97^{275}$$ has a units digit same as that of 7 raised to the remainder of $$\frac{275}{4}$$ which is $$7^3$$=343

Therefore $$97^{275}$$ has a units digit of $$3$$

Similarly $$32^{44}$$ has a units digit $$6$$ since cyclicity of $$2$$ is $$4$$

Therefore the difference is xxxxxxx3 - yyyyyyy6 = zzzzz7

Originally posted by firas92 on 02 Jul 2019, 08:06.
Last edited by firas92 on 02 Jul 2019, 09:39, edited 5 times 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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Updated on: 02 Jul 2019, 22:26
2

97^275 - 32^44

7^1=7
7^2=9
7^3=3
7^4=1
7^5=7 .. the trend continues 7,9,3,1

2^1=2
2^2=4
2^3=8
2^4=6
2^5=2 .. the trend continues 2,4,8,6

97^275 - 32^44

Effectively 7^275 - 2^44
7^275 will have the same last digit as 7^3
2^44 will have the same last digit as 2^4

7^3-2^4

3 - 6
(Same as 13 - 6 carried over from the tens digfit)
= 7

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Originally posted by prashanths on 02 Jul 2019, 08:25.
Last edited by prashanths on 02 Jul 2019, 22:26, edited 3 times 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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02 Jul 2019, 08:32
2
Quote:
What is the digit on the units place in the expanded value of 97275–324497275–3244?

A. 1
B. 3
C. 5
D. 7
E. 9

in no. 97, cyclicity of unit's digit 7 is 7,9,3,1
and 275 /4 gives 3 as a remainder.
in no.32, cyclicity of unit's digit 2 is 2,4,8,6
and 44/4 gives 0 as remainder.

hence the equation is ...3-...6 which will end up to be 7 as the unit's digit.
hence 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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02 Jul 2019, 22:16
2
Unit digit of 97^275 would be same as 7^275, which would be same as 7^3 (the unit digits of 7^x repeat after x = 4), which is 3.
Similarly, unit digit of 32^44 would be same as 2^44, which would be same as 2^4 (the unit digits of 2^x repeat after x = 4), which is 6.

Now, ____3-____6 will be same as 13-6 = 7 (where "____x" represents some number string ending with x)
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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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03 Jul 2019, 06:34
2
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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Re: What is the digit on the units place in the expanded value of 97^275 –  [#permalink]

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02 Jul 2019, 08:12
1
We have to find unit digit of 97^275–32^44
275=3 mod 4
Hence unit digit of 97^275 is same as 7^3=3

44=0 mod 4
Hence unit digit of 32^44 is same as 2^4=6

unit digit of 97^275–32^44= x3-y6=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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02 Jul 2019, 08:14
1
Cyclicity of last digit of 7= {7, 9,3,1,7.......}
Cyclicity of last digit of 2= {2,4,8,6,2........}

so every power of form 4K+1 has same last digit.
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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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02 Jul 2019, 08:15
1
1
Cyclicity of 7= 4. Notice the unit's digit-
7^1=7,
7^2=9
7^3=3
7^4=1
7^5=7 (repeats)

Remainder when 275 divided by 4 = 3
Therefore Unit's digit of 97^275=> 7^3 => 3

Likewise, cyclicity of 2=4
And Remainder when 44 divided by 4 = 0
Therefore, Unit's digit of 32^44 => 2^4 => 6

Difference 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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02 Jul 2019, 08:18
1
use cyclicity to solve
so
97^275 ; unit digit ; 3
and 32^44; unit digit ; 6
∆ ; 3-6 ; 7
IMO D

What is the digit on the units place in the expanded value of 97275–324497275–3244?

A. 1
B. 3
C. 5
D. 7
E. 9
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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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02 Jul 2019, 08:20
1
Based on cyclic properties of units digit 7's and 2's unit digit repeats itself in ^4. so 7^275 units digit similar to 7^3 and 2^44 will be similar to 2^4. 343-16 gives unit digit of 7. IMO 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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02 Jul 2019, 08:21
1
Units place for 7^275 = 3
Units place for 2^44 = 6

Now 3-6 = -3. We need to add 10 to any negative difference in such questions to get the actual value of units place.

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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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02 Jul 2019, 08:21
1
IMO D

7 has cyclicity of 4, that is 7*7*7*7 has unit's digit as 1
2 has cyclicity of 4, that is 2*2*2*2 has unit's digit as 6

Now we just have to reorder the given expression
97^275–32^44 = ((97)^(4*68)) * (97^3) - 32^(4*11) => 7^3 has unit's digit 3 - 2^(4*11) has unit's digit 6 => answer is 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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02 Jul 2019, 08:22
1
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

In 97^275, unit digit will depend on 7^275 and in 32^44 will depend on 2^44.
Unit digit of
7^1 = 7
7^2 = 9
7^3 = 3
7^4 = 1
7^5 = 7
In general unit digit of 7^(4x)=1
275=4*68+3
Unit digit of 97^275 = 3

Similarily,
Unit digit of
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 6
2^5 = 2
In general unit digit of 2^(4x)=6
44=4*11
Unit digit of 32^44 = 6

Unit digit of 97^275-32^44 = 13- 6 = 7 since 3-6 gives negative value and one is borrowed from 10's digit.

IMO 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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02 Jul 2019, 08:22
1
Concept : Cyclicity

97^275 - 32^44
Unit digit of 97 is same as 7 and 32 is same as 2
--> 97^275 - 32^44
--> 7^275 - 2^44
Cyclicity of 7 is 4
--> 275 = 4*68 + 3
--> 7^275 = 7^3 = 343
Unit Digit = 3

Cyclicity of 2 is 4
--> 44 = 4*10 + 4
--> 2^44 = 2^4 = 16
Unit Digit = 6

So, Unit Digit of 7^275 - 2^44 = 3 - 6
= 13 - 6
= 7

IMO Option 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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02 Jul 2019, 08:23
1
My answer is (D), i.e. 7. The concept is pretty straightforward, but it can be time consuming. I just need to speed up.

97^275 is clearly larger than 32^44. So after the subtraction, the result is positive. We just need to know the unit digit for both 97^275 and 32^44.

Notice
97 ^ 1 ending in 7
97 ^ 2 ending in 9
97 ^ 3 ending in 3
97 ^ 4 ending in 1
97 ^ 5 ending in 7, which is the same as 97 ^ 1. We already see a pattern here.
...
97 ^ 275 = 97 ^ (4 * 68 + 3) ending in 3.

Similarly, for 32 ^ 44, we can find out it ends in 6.

Because 13 - 6 = 7, I choose (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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02 Jul 2019, 08:25
1
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

power->unit digit
---------------
7^1 ->7
7^2->9
7^3 -> 3: 275 = 4* 68 + 3
7^4->1
7^5->7 >> so repeating after 4th power: 7931-7931----

power->unit digit
---------------
2^1->2
2^2->4
2^3->8
2^4->6: 44 = 4*11 + 0
2^5->2>> so repeating after 4th power: 2486-2486---

97^275 >32^44, so 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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02 Jul 2019, 08:26
1
7 and 2 follow a cycle when it comes to the unit's place.
7^1= 7, 7^2= 49 , 7^3=343, 7^4= 2401, 7^5= 16807. Hence, the unit's digit has a cycle of 7,9,3,1. By finding the remainder when 275 is divided by 4, we can find the corresponding term in the series which gives us a remainder of 3. Hence, the unit's digit would be 3 for the first term.
Similarly, 2 follows the same cycle of 2,4,8,6. Hence, when 44 is divided by 4 it gives a remainder of 0, which corresponds to the last digit being 6 for the second term.
3-6 would then lead to a difference of 7 and that would be the answer. Hence, option D
Re: What is the digit on the units place in the expanded value of 97^275 –   [#permalink] 02 Jul 2019, 08:26

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