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What is the tens digit of 6^17? (A) 1 (B) 3 (C) 5 (D) 7 (E) 9

There are several ways to deal with this problems some easier some harder, but almost all of them are based on the pattern recognition.

The tens digit of 6 in integer power starting from 2 (6^1 has no tens digit) repeats in pattern of 5: {3, 1, 9, 7, 5}: The tens digit of 6^2=36 is 3; The tens digit of 6^3=216 is 1; The tens digit of 6^4=...96 is 9 (how to calculate: multiply 16 by 6 to get ...96 as the last two digits); The tens digit of 6^5=...76 is 7 (how to calculate: multiply 96 by 6 to get ...76 as the last two digit); The tens digit of 6^6=...56 is 5 (how to calculate: multiply 76 by 6 to get ...56 as the last two digits); The tens digit of 6^7=...36 is 3 again (how to calculate: multiply 56 by 6 to get ...36 as the last two digits).

Hence, 6^2, 6^7, 6^12, 6^17, 6^22, ... will have the same tens digit of 3.

Re: What is the tens digit of 6^17? [#permalink]
10 Feb 2012, 04:10

1

This post received KUDOS

well, this question demands calculation to see a pattern of tens digits keep calculating till it's confirmed that u have hit a pattern. 6^1 = 6 6^2 = 36 6^3 = 216 now don't multiply 216 by 6, rather we are interested in only first two digits to know the outcome so 6^4 = 96 ( 16 x 6) 6^5 = 576 ( 96 x 6) 6^ 6 = 456 ( 76 x 6) 7^ 6 =336 ( 56 x 6) so now we have the pattern in tens digit i.e. 3 in (6^2), 1 in (6^3), 9 in (6^4), 7 in (6^5), 5 in (6^6), 3 in (6^7),

so the tens digit is 3 for the 2,7,12 and 17 times.. IMO B _________________

Fire the final bullet only when you are constantly hitting the Bull's eye, till then KEEP PRACTICING.

Note that when you multiply, you don't have to finish it all the way, knowing the tens digit should suffice.... Also, using the table we have we can calculate 6^{10} and 6^{17}. We work with what we already have above/

Re: What is the tens digit of 6^17? [#permalink]
13 Jan 2013, 12:15

bunuel would you please post me a link on the topic of exponents and powers from gmat math book if it has been finished..i want to learn and master way u hav solved the problem

Re: What is the tens digit of 6^17? [#permalink]
14 Jan 2013, 00:44

Expert's post

chiccufrazer1 wrote:

bunuel would you please post me a link on the topic of exponents and powers from gmat math book if it has been finished..i want to learn and master way u hav solved the problem

What is the tens digit of 6^17? (A) 1 (B) 3 (C) 5 (D) 7 (E) 9

There are several ways to deal with this problems some easier some harder, but almost all of them are based on the pattern recognition.

The tens digit of 6 in integer power starting from 2 (6^1 has no tens digit) repeats in pattern of 5: {3, 1, 9, 7, 5}: The tens digit of 6^2=36 is 3; The tens digit of 6^3=216 is 1; The tens digit of 6^4=...96 is 9 (how to calculate: multiply 16 by 6 to get ...96 as the last two digits); The tens digit of 6^5=...76 is 7 (how to calculate: multiply 96 by 6 to get ...76 as the last two digit); The tens digit of 6^6=...56 is 5 (how to calculate: multiply 76 by 6 to get ...56 as the last two digits); The tens digit of 6^7=...36 is 3 again (how to calculate: multiply 56 by 6 to get ...36 as the last two digits).

Hence, 6^2, 6^7, 6^12, 6^17, 6^22, ... will have the same tens digit of 3.

Answer: B.

i have noticed that every number has 6 as the unit digit..is it the same for other numbers that they repeat each of the unit's digit throughout when it is being raised to powers of consecutive integers

What is the tens digit of 6^17? (A) 1 (B) 3 (C) 5 (D) 7 (E) 9

There are several ways to deal with this problems some easier some harder, but almost all of them are based on the pattern recognition.

The tens digit of 6 in integer power starting from 2 (6^1 has no tens digit) repeats in pattern of 5: {3, 1, 9, 7, 5}: The tens digit of 6^2=36 is 3; The tens digit of 6^3=216 is 1; The tens digit of 6^4=...96 is 9 (how to calculate: multiply 16 by 6 to get ...96 as the last two digits); The tens digit of 6^5=...76 is 7 (how to calculate: multiply 96 by 6 to get ...76 as the last two digit); The tens digit of 6^6=...56 is 5 (how to calculate: multiply 76 by 6 to get ...56 as the last two digits); The tens digit of 6^7=...36 is 3 again (how to calculate: multiply 56 by 6 to get ...36 as the last two digits).

Hence, 6^2, 6^7, 6^12, 6^17, 6^22, ... will have the same tens digit of 3.

Answer: B.

i have noticed that every number has 6 as the unit digit..is it the same for other numbers that they repeat each of the unit's digit throughout when it is being raised to powers of consecutive integers

Posted from my mobile device

No. You could test that very easily yourself. Is the units digit of 2^2 equal 2? No, its 4.

• Integer ending with 0, 1, 5 or 6, in the integer power k>0, has the same last digit as the base. • Integers ending with 2, 3, 7 and 8 have a cyclicity of 4.