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You are watching: What is the units digit of 3^43

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If n = (33)^43 + (43)^33 what is the units digit the n?A. 0B. 2C. 4D. 6E. 8The OA is A.How have the right to I know the systems digit without making the whole calculation? I need some help. Please.
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GMATGuruNYGMAT Instructor Posts: 15533Joined: 25 might 2010Location: new York, NY Thanked: 13060 time Followed by:1900 members GMAT Score:790
3¹ --> units digit the 3. 3² --> systems digit of 9. (Since the product the the coming before units digit and also 3 = 3*3 = 9.) 3³ --> systems digit the 7. (Since the product of the coming before units digit and also 3 = 9*3 = 27.) 3� --> units digit the 1. (Since the product the the coming before units digit and 3 = 7*3 = 21.) from here, the devices digits will certainly repeat in the very same pattern: 3, 9, 7, 1. The units digit repeat in a cycle OF 4. Implication: as soon as an integer with a systems digit of 3 is elevated to a strength that is a lot of of 4, the devices digit will certainly be 1. Thus:33�� and also 43³² each have actually a units digit of 1. From here, the bike of systems digits will certainly repeat: 3, 9, 7, 1...Thus:33�¹ and also 43³³ each have a units digit the 3.33�² has actually a units digit of 9.33�³ has actually a devices digit of 7.Result:Since n = 33�³ + 43³³, n ---> (units number of 3) + (units number of 7) = devices digit that 0.The exactly answer is A.
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GMAT Instructor Posts: 1460Joined: 09 Apr 2015Location: new York, NY Thanked: 39 time Followed by:21 members
Since we space asked to recognize only the units digit that n, we can rewrite the expression as:n = 3^43 + 3^33Let"s now recognize the units digit that 3^43 and also 3^33Let"s begin by analyzing the sample of the systems digits of 3^n for confident integer values of n. The is, let"s look in ~ the sample of the devices digits of powers of 3. When writing the end the pattern, notification that we are ONLY concerned with the devices digit that 3 elevated to every power. 3^1 = 3 3^2 = 9 3^3 = 7 3^4 = 1 3^5 = 3The pattern of the devices digit of strength of 3 repeats every 4 exponents. The pattern is 3-9-7-1. In this pattern, all optimistic exponents that are multiples that 4 will create a 1 as its units digit. Thus: 3^44 has a systems digit that 1.Therefore, 3^43 has actually a units digit the 7.and3^32 has actually a units digit of 1.Therefore, 3^33 has a units digit of 3.Thus, n has actually a systems digit that 0, since the systems digit that 7 + 3 = 10 is 0.Answer: A
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As Mitch described, finding the pattern in units digits is the crucial to addressing this problem. For an ext practice top top this concept, see:https://www.stclairdrake.net/what-is-the ... Tml#554073https://www.stclairdrake.net/what-is-the ... Tml#544267https://www.stclairdrake.net/what-is-the ... Tml#800962https://www.stclairdrake.net/if-n-and-a- ... Tml#784629