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# What is the units digit of (3^{101})(7^{103})?

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
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What is the units digit of (3^{101})(7^{103})?  [#permalink]

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29 Aug 2018, 01:56
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25% (medium)

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[Math Revolution GMAT math practice question]

What is the units digit of $$(3^{101})(7^{103})$$?

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

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"Only $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Manager Joined: 14 Jun 2018 Posts: 222 Re: What is the units digit of (3^{101})(7^{103})? [#permalink] ### Show Tags 29 Aug 2018, 02:04 3^101 * 7^103 = 21^101 * 49 cycle of last digit of 21 is 1. Ans E Director Joined: 20 Feb 2015 Posts: 788 Concentration: Strategy, General Management Re: What is the units digit of (3^{101})(7^{103})? [#permalink] ### Show Tags 29 Aug 2018, 02:06 MathRevolution wrote: [Math Revolution GMAT math practice question] What is the units digit of $$(3^{101})(7^{103})$$? $$A. 1$$ $$B. 3$$ $$C. 5$$ $$D. 7$$ $$E. 9$$ $$(3^{101})(7^{103})$$ 101= 4k+1 103=4k+3 above can be written as $$(3^{1})(7^{3})$$ 3*343 units digit = 9 Senior Manager Joined: 04 Aug 2010 Posts: 447 Schools: Dartmouth College Re: What is the units digit of (3^{101})(7^{103})? [#permalink] ### Show Tags 29 Aug 2018, 03:38 MathRevolution wrote: [Math Revolution GMAT math practice question] What is the units digit of $$(3^{101})(7^{103})$$? $$A. 1$$ $$B. 3$$ $$C. 5$$ $$D. 7$$ $$E. 9$$ When an integer is raised to consecutive powers, the resulting units digits repeat in a CYCLE. $$3^{101}$$: 3¹ --> units digit of 3. 3² --> units digit of 9. (Since the product of the preceding units digit and 3 = 3*3 = 9.) 3³ --> units digit of 7. (Since the product of the preceding units digit and 3 = 9*3 = 27.) 3⁴ --> units digit of 1. (Since the product of the preceding units digit and 3 = 7*3 = 21.) From here, the units digits will repeat in the same pattern: 3, 9, 7, 1. The units digit repeat in a CYCLE OF 4. Implication: When an integer with a units digit of 3 is raised to a power that is a multiple of 4, the units digit will be 1. Thus, $$3^{100}$$has a units digit of 1. From here, the cycle of units digits will repeat: 3, 9, 7, 1... Thus, $$3^{101}$$ has a units digit of 3. 7¹⁰³: 7¹ --> units digit of 7. 7² --> units digit of 9. (Since the product of the preceding units digit and 7 = 7*7 = 49.) 7³ --> units digit of 3. (Since the product of the preceding units digit and 7 = 9*7 = 63.) 7⁴ --> units digit of 1. (Since the product of the preceding units digit and 7 = 3*7 = 21.) From here, the units digits will repeat in the same pattern: 7, 9, 3, 1. The units digit repeat in a CYCLE OF 4. Implication: When an integer with a units digit of 7 is raised to a power that is a multiple of 4, the units digit will be 1. Thus, $$7^{100}$$ has a units digit of 1. From here, the cycle of units digits will repeat: 7, 9, 3, 1... $$7^{101}$$--> units digit of 7. $$7^{102}$$ --> units digit of 9. $$7^{103}$$--> units digit of 3. Result: $$3^{101}7^{103}$$ = (integer with a units digit of 3)(integer with a units digit of 3) = integer with a units digit of 9. _________________ GMAT and GRE Tutor Over 1800 followers GMATGuruNY@gmail.com New York, NY If you find one of my posts helpful, please take a moment to click on the "Kudos" icon. Available for tutoring in NYC and long-distance. For more information, please email me at GMATGuruNY@gmail.com. Board of Directors Status: QA & VA Forum Moderator Joined: 11 Jun 2011 Posts: 4570 Location: India GPA: 3.5 WE: Business Development (Commercial Banking) Re: What is the units digit of (3^{101})(7^{103})? [#permalink] ### Show Tags 29 Aug 2018, 07:40 1 MathRevolution wrote: [Math Revolution GMAT math practice question] What is the units digit of $$(3^{101})(7^{103})$$? $$A. 1$$ $$B. 3$$ $$C. 5$$ $$D. 7$$ $$E. 9$$ $$3^4$$ = Units digit 1 $$7^4$$ = Units digit 1 $$(3^{101})(7^{103})$$ = $$(3^{4*25}*3^1)(7^{4*25}*7^3)$$ 3^1 will have units digit 1 and 7^3 will have units digit 3 So, The units digit of the expression will be $$1*3*1*3 = 9$$ , Answer must be (E) _________________ Thanks and Regards Abhishek.... PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS How to use Search Function in GMAT Club | Rules for Posting in QA forum | Writing Mathematical Formulas |Rules for Posting in VA forum | Request Expert's Reply ( VA Forum Only ) VP Joined: 09 Mar 2016 Posts: 1258 Re: What is the units digit of (3^{101})(7^{103})? [#permalink] ### Show Tags 29 Aug 2018, 08:48 Abhishek009 wrote: MathRevolution wrote: [Math Revolution GMAT math practice question] What is the units digit of $$(3^{101})(7^{103})$$? $$A. 1$$ $$B. 3$$ $$C. 5$$ $$D. 7$$ $$E. 9$$ $$3^4$$ = Units digit 1 $$7^4$$ = Units digit 1 $$(3^{101})(7^{103})$$ = $$(3^{4*25}*3^1)(7^{4*25}*7^3)$$ 3^1 will have units digit 1 and 7^3 will have units digit 3 So, The units digit of the expression will be $$1*3*1*3 = 9$$ , Answer must be (E) hey there Abhishek009 hope your solo guitar career is thriving let me know when you are gving your next rock concert , i will buy tickets if 3^1 will have units digit 1 and 7^3 will have units digit 3 So we have unit digit 1 and unit digit 3 hence 1*3 = 3 ? where from did you get so many numbers $$1*3*1*3 = 9$$ have a great evening Board of Directors Status: QA & VA Forum Moderator Joined: 11 Jun 2011 Posts: 4570 Location: India GPA: 3.5 WE: Business Development (Commercial Banking) Re: What is the units digit of (3^{101})(7^{103})? [#permalink] ### Show Tags 29 Aug 2018, 09:06 1 dave13 wrote: hey there Abhishek009 hope your solo guitar career is thriving let me know when you are gving your next rock concert , i will buy tickets if 3^1 will have units digit 1 and 7^3 will have units digit 3 So we have unit digit 1 and unit digit 3 hence 1*3 = 3 ? where from did you get so many numbers $$1*3*1*3 = 9$$ have a great evening $$= (3^{4∗25}∗3^1)(7^{4∗25}∗7^3)$$ $$= (1^{25}∗3)(1^{25}∗3)$$ { Units digit of 3^4 = 1 and Units digit of 7^4 = 3 } $$= 1* 3 * 1 * 3$$ Hope this helps!!! PS : Mr Bean is my fav character, and I really love your innocent looking DP, good evening friend, plz feel free to revert in case of any further doubt... _________________ Thanks and Regards Abhishek.... PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS How to use Search Function in GMAT Club | Rules for Posting in QA forum | Writing Mathematical Formulas |Rules for Posting in VA forum | Request Expert's Reply ( VA Forum Only ) Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 7728 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: What is the units digit of (3^{101})(7^{103})? [#permalink] ### Show Tags 31 Aug 2018, 01:15 => The units digit is the remainder when $$(3^{101})(7^{103})$$ is divided by $$10$$. The remainders when powers of $$3$$ are divided by $$10$$ are $$3^1: 3,$$ $$3^2: 9,$$ $$3^3: 7,$$ $$3^4: 1,$$ $$3^5: 3,$$ So, the units digits of $$3^n$$ have period $$4$$: They form the cycle $$3 -> 9 -> 7 -> 1.$$ Thus, $$3^n$$ has the units digit of $$3$$ if $$n$$ has a remainder of $$1$$ when it is divided by $$4$$. The remainder when $$101$$ is divided by $$4$$ is $$1$$, so the units digit of $$3^{101}$$ is $$3$$. The remainders when powers of $$7$$ are divided by $$10$$ are $$7^1: 7,$$ $$7^2: 9,$$ $$7^3: 3,$$ $$7^4: 1,$$ $$7^5: 7,$$ So, the units digits of $$7^n$$ have period $$4$$: They form the cycle $$7 -> 9 -> 3 -> 1$$. Thus, $$7^n$$ has the units digit of $$3$$ if $$n$$ has a remainder of $$3$$ when it is divided by $$4$$. The remainder when $$103$$ is divided by $$4$$ is $$3$$, so the units digit of $$7^{103}$$ is $$3$$. Thus, the units digit of $$(3^{101})(7^{103})$$ is $$3*3 = 9.$$ Therefore, the answer is E. Answer: E _________________ 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$79 for 1 month Online Course"
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What is the units digit of (3^{101})(7^{103})?  [#permalink]

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Updated on: 03 Oct 2018, 10:34
MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the units digit of $$(3^{101})(7^{103})$$?

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

3^101 * 7^103

cycle of 3 is 3,9,7,1
101st through cyclicity will be 3
cycle of 7 is 7,9,3,1
103rd through cyclicity will be 3

Units digit i.e. 3*3 = 9

Ans (E)

Originally posted by Natty97 on 31 Aug 2018, 10:00.
Last edited by Natty97 on 03 Oct 2018, 10:34, edited 1 time in total.
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Re: What is the units digit of (3^{101})(7^{103})?  [#permalink]

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03 Sep 2018, 19:01
MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the units digit of $$(3^{101})(7^{103})$$?

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

Let’s start by evaluating the pattern of the units digits 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 concerned ONLY with the units digit of 3 raised to each power.

3^1 = 3

3^2 = 9

3^3 = 7

3^4 = 1

3^5 = 3

The pattern of the units digit 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 1 as its units digit. Thus:

3^100 has a units digit of 1, and so 3^101 has a units digit of 3.

Next, we can evaluate the pattern of the units digits of 7^n for positive integer values of n. That is, let’s look at the pattern of the units digits of powers of 7. When writing out the pattern, notice that we are concerned ONLY with the units digit of 7 raised to each power.

7^1 = 7

7^2 = 9

7^3 = 3

7^4 = 1

7^5 = 7

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

7^104 has a units digit of 1, and so 7^103 has a units digit of 3.

Thus, the units digit of 3^101 x 7^103 is 3 x 3 = 9.

Answer: E
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Re: What is the units digit of (3^{101})(7^{103})?  [#permalink]

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03 Oct 2018, 08:31
Top Contributor
MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the units digit of $$(3^{101})(7^{103})$$?

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

This is a great question for applying the following property: (x^n)(y^n) = (xy)^n
For example, (3^7)(5^7) = 15^7

(3^101)(7^103) = (3^101)(7^101)(7^2) [rewrote 7^103 as the product of 7^101 and 7^2]
= (3^101)(7^101)(49) [evaluated 7^2]
= (21^101)(49) [applied above property]

Notice that 21^n will have units digit 1 for all positive integer values of n
So, = 21^101 = --------1 [some big number ending in 1]
So, we get:
= (--------1)(49)
= --------9

Answer: E

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Re: What is the units digit of (3^{101})(7^{103})?   [#permalink] 03 Oct 2018, 08:31
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