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What is the units digit of 9^19 - 7^15?

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What is the units digit of 9^19 - 7^15? [#permalink]

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What is the units digit of 9^19 - 7^15?

a) 2
b) 4
c) 5
d) 6
e) 7
Spoiler: OA

Originally posted by alanforde800Maximus on 10 Oct 2016, 07:31.
Last edited by Bunuel on 10 Oct 2016, 07:33, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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Re: What is the units digit of 9^19 - 7^15? [#permalink]

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alanforde800Maximus wrote:
What is the units digit of 9^19 - 7^15?

a) 2
b) 4
c) 5
d) 6
e) 7


Answer D.

9^19 has the last digit of 9.( 9 has the cyclicity of 2 and the last digits get repeated as 9,1)

7^15 has the last digit of 3.( 7 has the cyclicity of 4 and the last digits get repeated as 7,9,3,1)

So, last digit of 1st - last digit of 2nd will give the last digit of the result.

or 9-3 =6 Answer.
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Re: What is the units digit of 9^19 - 7^15? [#permalink]

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alanforde800Maximus wrote:
What is the units digit of 9^19 - 7^15?

a) 2
b) 4
c) 5
d) 6
e) 7


\(9^{Odd}\) = Units digit 9
\(9^{Even}\) = Units digit 1

So, Units digit of \(9^{19}\) will be 9

Cyclycity of 7 is 4

So, \(7^{15}\) = \(7^{12}\) x \(7^3\)

\(7^{12}\) will have units digit 1
\(7^{3}\) will have units digit 3

So, The units digit of \(9^{19} - 7^{15}\) =>\(9 - 1*3 =6\)

Hence answer will be (D) 3
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Re: What is the units digit of 9^19 - 7^15? [#permalink]

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New post 31 Aug 2017, 19:18
alanforde800Maximus wrote:
What is the units digit of 9^19 - 7^15?

a) 2
b) 4
c) 5
d) 6
e) 7


For cycle questions such as these you simply use division to find the units digit

9 occurs in cycles of 2: 9,1 so you divide the exponent- 19/2 - whatever the remainder is will be the answer - so in this case the remainder is 1 so the units digit will be the first digit in the series "9." When there is no remainder, then the units digit is always the last number in the series. Always start with the number it self- so in a cycle of 7 start with 7 ,9 , 3 1 or in a cycle of 9 you go 9,1,9,1

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Re: What is the units digit of 9^19 - 7^15? [#permalink]

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New post 31 Aug 2017, 22:24
Number---^1---^2---^3---^4------Cyclicity
2----------- 2-----4------8-----6----------4
3----------- 3-----9------7-----1----------4
4----------- 4-----6------4-----6----------2
5----------- 5-----5------5-----5----------1
6----------- 6-----6------6-----6----------1
7----------- 7-----9------3-----1----------4
8----------- 8-----4------2-----6----------4
9----------- 9-----1------9-----1----------2

The table above gives the cyclicity of any number.
If needed, memorize this table for numbers 2,3,7,8 which have the cyclicity of 4.

However, if we need to derive the cyclicity for number 9 and 7
we can do so by this method

For this problem
Units digit
\(9^odd\) = 9
\(9^even\) = 1
\(7^1\) = 7
\(7^2\) = 9
\(7^3\) = 3
\(7^4\) = 1

From the above table, we can clearly say that \(9^19 - 7^15\).
Since 9^19 has units digit 9 and 7^15 has units digit 3,
we will have units digit for the expression as 9-3(6)(Option D)
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Re: What is the units digit of 9^19 - 7^15?   [#permalink] 31 Aug 2017, 22:24
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