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WE 1: 7 Yrs in Automobile (Commercial Vehicle industry)

Re: Questions that require you to find last digit. [#permalink]

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21 Oct 2012, 20:33

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supreetb wrote:

Find the ones digit of 73^350

Answer choices : 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.

Hence Answer E.
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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.

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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.

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.

Answer: E
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