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What is the units digit of 43^19?

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What is the units digit of 43^19? [#permalink]

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New post 07 Apr 2017, 02:44
00:00
A
B
C
D
E

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Question Stats:

78% (00:18) correct 22% (00:33) wrong based on 87 sessions

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What is the units digit of 43^19? [#permalink]

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New post 07 Apr 2017, 05:04
3^1 = 3
3^2 = 9
3^3 =...7
3^4 =...1

Cycling through 19 times, it ends on 7. Answer "D".
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Last edited by emockus on 07 Apr 2017, 08:56, edited 1 time in total.
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Re: What is the units digit of 43^19? [#permalink]

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New post 07 Apr 2017, 05:50
Bunuel wrote:
What is the units digit of \(43^{19}\)?

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



3^1 = 3
3^2 = 9
3^3 = 7
3^4 = 1 End of pattern. The 4th, 8th, 16th, 20th power. 19th power will be one less. (D), 7.

3^5 = 3
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Re: What is the units digit of 43^19? [#permalink]

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New post 07 Apr 2017, 07:34
What is the units digit of 43^19?

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


3 has a cyclicity of 4
19=4q+3
so basically 3^3=7 unit digit
answer is D
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What is the units digit of 43^19? [#permalink]

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New post 07 Apr 2017, 08:02
Bunuel wrote:
What is the units digit of \(43^{19}\)?

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



\(\begin{align}
43^{19} &=43^{18+1} \\
&=(43^2)^9 \times 43 \\
&= (...9)^{8+1} \times 43 \\
&= (...9)^{2 \times 4} \times [(...9) \times 43] \\
&= [(...9)^2]^4 \times (...7) \\
&=(...1)^4 \times (...7) \\
&= (...1) \times (...7) \\
&= (...7)
\end{align}\)

The answer is D
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Last edited by broall on 09 Apr 2017, 19:11, edited 1 time in total.
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What is the units digit of 43^19? [#permalink]

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New post 07 Apr 2017, 08:53
Once you identify the pattern (3,9,7,1) you can set up a grid:



3: 1,5, 9,13,17
9: 2,6,10,14,18
7: 3,7,11,15,19
1: 4,8/12,16
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Re: What is the units digit of 43^19? [#permalink]

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New post 07 Apr 2017, 09:07
Bunuel wrote:
What is the units digit of \(43^{19}\)?

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


units digit cycle from 3^1 to 3^4=3,9,7,1
19/4 gives a remainder of 3
3rd digit in cycle is 7
D
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Re: What is the units digit of 43^19? [#permalink]

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New post 07 Apr 2017, 09:14
the units digit is same as that of 3^19

3 has a cyclicity of 4..hence units digit of 3^3 = 7

ans 7
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Re: What is the units digit of 43^19? [#permalink]

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New post 08 Apr 2017, 11:49
3^0 = 1
3^1 =3
3^2 = 9
3^3=7
3^4 = 1

20 is a multiple of 4. 3^20 = 1
Therefore 3^19 = 7

Which is answer D
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Re: What is the units digit of 43^19? [#permalink]

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New post 14 Apr 2017, 04:41
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Expert's post
Bunuel wrote:
What is the units digit of \(43^{19}\)?

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


Since we need to determine the units digit of 43^19, we really are determining the units digit of 3^19.

The pattern of units digits of 3 when raised to a positive integer exponent is:

3^1 = 3

3^2 = 9

3^3 = 7

3^4 = 1

3^5 = 3

We see that the pattern is 3-9-7-1. We can generalize this pattern by saying that when the exponent is a multiple of 4, the units digit will be 1. For example, both 3^16 and 3^20 have a units digit of 1.
The units digit of 3^20 is 1, and so 3^19 has a units digit of 7.

Answer: D
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Re: What is the units digit of 43^19?   [#permalink] 14 Apr 2017, 04:41
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