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# What is the units digit of 248^20?

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Math Revolution GMAT Instructor
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What is the units digit of 248^20?  [#permalink]

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07 Jul 2017, 01:11
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68% (00:37) correct 32% (00:35) wrong based on 119 sessions

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What is the units digit of $$248^{20}$$?

A. 0
B. 2
C. 4
D. 6
E. 8

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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Senior Manager Joined: 14 Oct 2015 Posts: 250 GPA: 3.57 What is the units digit of 248^20? [#permalink] ### Show Tags Updated on: 07 Jul 2017, 02:39 MathRevolution wrote: What is the units digit of $$248^{20}$$? A. 0 B. 2 C. 4 D. 6 E. 8 D factoring, you get 248 = 2 x 2 x 2 x 31 or (2^3 * 31)^20 This equals $$2^{60} * 31^{20}$$ We individually calculate units digits of the two factors. For 2: $$2^1 = 2$$ $$2^2 = 4$$ $$2^3 = 8$$ $$2^4 = 16$$ $$2^5 = 2$$ and the pattern of unit digits 2,4,8 and 6 repeats after every multiplication of 2. so $$2^{60}$$ has the units digit the same as $$2^4$$ which is 6. For 31: Units digit of 31x31 is 1 and it is again multiplied by 31 to result in units digit of 1 and the pattern continues. so essentially we end up with units digits of 6 and 1 for expressions of 2 and 31. Multiplying both results in 6 which would be the units digit of $$248^{20}$$ _________________ Please hit Kudos if this post helped you inch closer to your GMAT goal. Procrastination is the termite constantly trying to eat your GMAT tree from the inside. There is one fix to every problem, working harder! Originally posted by jedit on 07 Jul 2017, 02:28. Last edited by jedit on 07 Jul 2017, 02:39, edited 2 times in total. Manager Joined: 13 Aug 2015 Posts: 211 GMAT 1: 710 Q49 V38 GPA: 3.82 WE: Corporate Finance (Retail Banking) Re: What is the units digit of 248^20? [#permalink] ### Show Tags 07 Jul 2017, 02:35 1 When numbers have last digit of 2, 3, 7, 8, they have a cyclicity of 4. When 20 is divided by 4, the remainder is 0. So, 8^4 has unit digit of 6. HENCE, ANS IS D _________________ If you like my posts, please give kudos. Help me unlock gmatclub tests. Director Joined: 04 Dec 2015 Posts: 700 Location: India Concentration: Technology, Strategy Schools: ISB '19, IIMA , IIMB, XLRI WE: Information Technology (Consulting) What is the units digit of 248^20? [#permalink] ### Show Tags 07 Jul 2017, 06:18 1 MathRevolution wrote: What is the units digit of $$248^{20}$$? A. 0 B. 2 C. 4 D. 6 E. 8 $$248^{20}$$ We just need to check the unit digit of $$8$$. $$8^1 = 8$$ ---------(Unit digit is $$8$$) $$8^2 = 64$$ -------- (Unit digit is $$4$$) $$8^3 = 64 * 8 = 4*8 = 32$$ ----- (we just need to multiply the unit digits to get the unit digit value, hence $$4*8$$) ----- (Unit digit is $$2$$) $$8^4 = 32 * 8 = 2*8 = 16$$ ----- (we just need to multiply the unit digits to get the unit digit value, hence $$2*8$$) ----- (Unit digit is $$6$$) $$8^5 = 16 * 8 = 6*8 = 48$$ ----- (we just need to multiply the unit digits to get the unit digit value, hence $$6*8$$) ------(Unit digit is $$8$$) $$8$$ has cyclicity of $$4$$. Hence $$248$$ will also have the cyclicity of $$4$$. $$20$$ is divisible by $$4$$, hence the remainder would be $$0$$. Hence the unit digit of $$248 = 6$$ Answer (D)... _________________ Please Press "+1 Kudos" to appreciate. Manager Joined: 13 Aug 2015 Posts: 211 GMAT 1: 710 Q49 V38 GPA: 3.82 WE: Corporate Finance (Retail Banking) What is the units digit of 248^20? [#permalink] ### Show Tags 07 Jul 2017, 06:50 MathRevolution wrote: What is the units digit of $$248^{20}$$? A. 0 B. 2 C. 4 D. 6 E. 8 Theory: The cyclicity of 2,3,7,8 is 4. When Power is divided by cyclicity; Remainder is zero, then unit digit^cyclicity When power is divided by cyclicity; Remainder is Non zero, then unit digit ^Remainder 20=4x5 + 0 Thus, remainder =0 so, last digit of 248^cyclicity(4) =8^4 =6 Ans is D _________________ If you like my posts, please give kudos. Help me unlock gmatclub tests. Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 5876 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: What is the units digit of 248^20? [#permalink] ### Show Tags 09 Jul 2017, 18:30 ==>You get $$~8^1=~8, ~8^2=~4, ~8^3=~2, ~8^4=~6,$$ so the units digit becomes 8-->4-->2-->6-->8, and the index has a period of 4. Thus, you get $$248^{20}=(248^4)^5=(~6)^5=~6.$$ The answer is D. Answer: D _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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Re: What is the units digit of 248^20?  [#permalink]

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12 Jul 2017, 16:48
MathRevolution wrote:
What is the units digit of $$248^{20}$$?

A. 0
B. 2
C. 4
D. 6
E. 8

Since we only care about units digits, we can rewrite the expression as:

8^20

Let’s start by evaluating the pattern of the units digits of 8^n for positive integer values of n. That is, let’s look at the pattern of the units digits of powers of 8. When writing out the pattern, notice that we are ONLY concerned with the units digit of 8 raised to each power.

8^1 = 8

8^2 = 4

8^3 = 2

8^4 = 6

8^5 = 8

The pattern of the units digit of powers of 8 repeats every 4 exponents. The pattern is 8–4–2–6. In this pattern, all positive exponents that are multiples of 4 will produce an 8 as the units digit. Thus:

8^20 has a units digit of 6.

Answer: D
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Re: What is the units digit of 248^20?  [#permalink]

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22 Sep 2017, 14:24
Top Contributor
MathRevolution wrote:
What is the units digit of $$248^{20}$$?

A. 0
B. 2
C. 4
D. 6
E. 8

Look for a pattern

248^1 = 248
248^2 = (248)(248) = ---4 [aside: we need not determine the other digits. All we care about is the units digit]
248^3 = (248)(248^2) = (248)(---4) = ----2
248^4 = (248)(248^3) = (248)(---2) = ----6
248^5 = (248)(248^4) = (248)(---6) = ----8

NOTICE that we're back to where we started.
248^5 has units digit 8, and 248^1 has units digit 8
So, at this point, our pattern of units digits keep repeating 8, 4, 2, 6, 8, 4, 2, 6, 8,...
We say that we have a "cycle" of 4, which means the digits repeat every 4 powers.

So, we get:
248^1 = --8
248^2 = ---4
248^3 = ----2
248^4 = ----6
248^5 = ----8
248^6 = ---4
248^7 = ----2
248^8 = ----6
248^9 = ----8
248^10 = ----4
etc.

Notice that when the exponent is a MULTIPLE of 4 (4, 8, 12, 16, ...), the units digit will be 6
Since 20 is a MULTIPLE of 4, we know that the units digit of 248^20 will be 6
Answer:

Here's an article I wrote on this topic (with additional practice questions): https://www.gmatprepnow.com/articles/un ... big-powers

Cheers,
Brent
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Re: What is the units digit of 248^20? &nbs [#permalink] 22 Sep 2017, 14:24
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# What is the units digit of 248^20?

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