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Intern  Status: Break IT
Joined: 12 Jun 2011
Posts: 3
Location: USA
Schools: ISB
WE 1: 4
Find the ones digit of 73^350  [#permalink]

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Difficulty:   5% (low)

Question Stats: 86% (00:47) correct 14% (01:21) wrong based on 657 sessions

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Find the ones digit of 73^350

A) 3
B) 5
C) 6
D) 7
E) 9

My answer was A, but it turned out incorrect. Can someone help understand why.

Originally posted by supreetb on 21 Oct 2012, 17:59.
Last edited by Bunuel on 22 Oct 2012, 08:20, edited 1 time in total.
Renamed the topic.
Senior Manager  Joined: 15 Jun 2010
Posts: 276
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Re: Questions that require you to find last digit.  [#permalink]

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3
2
supreetb wrote:
Find the ones digit of 73^350

A) 3
B) 5
C) 6
D) 7
E) 9

The source is grockit.

My answer was A, but it turned out incorrect. Can someone help understand why.

Cyclicity of 3 is 3,9,7,1 & after 4 multiplication again the cycle repeats.
So divide 350 by 4 and we get 87 as quotient and 2 as remainder. So cycle will run for 87 times and then 2 times more. So pick up the 2nd item from the cycle.

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Math Expert V
Joined: 02 Sep 2009
Posts: 58435
Re: Find the ones digit of 73^350  [#permalink]

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1
1
supreetb wrote:
Find the ones digit of 73^350

A) 3
B) 5
C) 6
D) 7
E) 9

My answer was A, but it turned out incorrect. Can someone help understand why.

The GMAT wording would be:

What is the units digit of 73^350?

The units digit of 73^350 will be the same as the units digit of 3^350.

3^1=3 --> the units digit is 3;
3^2=9 --> the units digit is 9;
3^3=27 --> the units digit is 7;
3^4=81 --> the units digit is 1;
3^5=243 --> the units digit is 3 AGAIN;
...

So, as you can see the units digit repeats in blocks of 4: {3, 9, 7, 1}, {3, 9, 7, 1}, ... Now, since 350=348+2=(multiple of 4)+2, then the units digit of 3^350 will be the second number in the pattern thus 9.

For more please check Number Theory chapter of Math Book: math-number-theory-88376.html

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Math Expert V
Joined: 02 Sep 2009
Posts: 58435
Re: Find the ones digit of 73^350  [#permalink]

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Bumping for review and further discussion.

For more on this kind of questions check Units digits, exponents, remainders problems collection.
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Intern  Joined: 20 Jan 2016
Posts: 43
GMAT 1: 600 Q47 V26 Re: Find the ones digit of 73^350  [#permalink]

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1
3,9,7,1 repeats after every four times

therefore 350/4 = 87 + 2 remainder

9 is the ones digit

so E is coorect

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Re: Find the ones digit of 73^350  [#permalink]

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supreetb wrote:
Find the ones digit of 73^350

A) 3
B) 5
C) 6
D) 7
E) 9

My answer was A, but it turned out incorrect. Can someone help understand why.

To find unit digit of $$73^{350}$$ , we need to find unit digit of $$3^{350}$$

Power of $$3$$ has cyclicity of $$4$$ $$(3,9,7,1)$$. $$\frac{350}{4}$$ gives remainder of $$2$$.

Therefore new power will be $$3^2$$. And Unit digit of $$3^2 = 9$$

Therefore Unit digit of $$73^{350} = 9$$

Manager  G
Joined: 12 Feb 2017
Posts: 70
Re: Find the ones digit of 73^350  [#permalink]

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The unit digit of 73^350 is same as that of 3^350.
Power of 3 has cyclicity of 4. That means set of 4 values repeats {3,9,7,1}.
3^1=3 (3)
3^2=9 (9)
3^3=27 (7)
3^4=81 (1)
3^5=243 (3)
and cycle continues.

Hence 3^350= 3^348 * 3^2
=3^2
=9.

Hence Option E.

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Target Test Prep Representative G
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Joined: 04 Mar 2011
Posts: 2815
Re: Find the ones digit of 73^350  [#permalink]

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supreetb wrote:
Find the ones digit of 73^350

A) 3
B) 5
C) 6
D) 7
E) 9

Since we only care about units digits, we really are determining the units digit of 3^350.

Let’s evaluate the units digit of 3^n for positive integer values of n. That is, let’s look at the pattern of the units digits of powers of 3. When writing out the pattern, notice that we are ONLY concerned with the units digit of 3 raised to a power.

3^1 = 3

3^2 = 9

3^3 = 7

3^4 = 1

3^5 = 3

The pattern of the units digits of powers of 3 repeats every 4 exponents. The pattern is 3–9–7–1. In this pattern, all positive exponents that are multiples of 4 will produce a 1 as the units digit. Thus:

3^348 has a units digit of 1 and 3^349 has a units digit of 3, and thus 3^350 has a units digit of 9.

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Re: Find the ones digit of 73^350  [#permalink]

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Hello from the GMAT Club BumpBot!

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_________________ Re: Find the ones digit of 73^350   [#permalink] 11 Oct 2018, 07:48
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