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If x is a positive integer, what is the units digit of [#permalink]
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11 Jul 2006, 10:32
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If x is a positive integer, what is the units digit of \((24)^{(2x + 1)}*(33)^{(x + 1)}*(17)^{(x + 2)}*(9)^{(2x)}\) ? A. 4 B. 6 C. 7 D. 8 E. 9
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Re: If x is a positive integer, what is the units digit of [#permalink]
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11 Jul 2006, 11:26
The answer is D
(24)^(2x + 1)
(2x+1) is an odd number since even + odd = odd
4 to an even power ends with a 4.
(33)^(x + 1)*(17)^(x + 2)
x could be even or odd. So if (x+1) is even, (x+2) will be odd. On the contrary, if (x+1) is odd, (x+2) will be even.
Trying some values for x:
If x =1
(33)^(2)*(17)^(3) = 9*3 = 27
If x= 2
(33)^(3)*(17)^(4) = 7*1 = 7
and so on......the table below shows the pattern:
Number
3 7
Power
2 9 9
3 7 3
4 1 1
5 3 7
6 9 9
(9)^(2x)
the power will be even, thus the units digit is 1.
Multiplying 4*7*1 = 8



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Re: If x is a positive integer, what is the units digit of [#permalink]
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11 Jul 2006, 11:44
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MA wrote: If x is a positive integer, what is the units digit of (24)^(2x + 1)*(33)^(x + 1)*(17)^(x + 2)*(9)^(2x)?
(A) 4 (B) 6 (C) 7 (D) 8 (E) 9
In these questions, since the answer will be true for any value of x, we can choose the min. value of x (in this case 1) and solve..
(D)



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Re: If x is a positive integer, what is the units digit of [#permalink]
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11 Jul 2006, 14:14
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Remember that in the GMAT, it pays to look for shortcuts!
33^(x+1)*17^(x+2)= ((33*17)^(x+1))*17 which has a units digit of 7!



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Re: If x is a positive integer, what is the units digit of [#permalink]
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Updated on: 11 Jul 2006, 20:31
good one. thanx kevin..
kevincan wrote: Remember that in the GMAT, it pays to look for shortcuts!
33^(x+1)*17^(x+2)= ((33*17)^(x+1))*17 which has a units digit of 7!
= (24)^(2x + 1) * (33)^(x + 1) * (17)^(x + 2) * (9)^(2x)
= 24 (336^x) * 33 (33^x) * 289 (17^x) * (81^x)
= (24 x 33 x 289) (336 x 33 x 17 x 81)^x
(24 x 33 x 289) has unit digit of 8
(336 x 33 x 17 x 81)^x has unit digit f 6 irrspective of value of x.
so (24 x 33 x 289) (336 x 33 x 17 x 81)^x has unit digit of 8.
Originally posted by MA on 11 Jul 2006, 19:39.
Last edited by MA on 11 Jul 2006, 20:31, edited 2 times in total.



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Re: If x is a positive integer, what is the units digit of [#permalink]
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30 Jun 2012, 23:45
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Hi,
\((24)^{2x + 1}*(33)^{x + 1}*(17)^{x + 2}*(9)^{2x}\) \(=(24^2)^x*24*33^x*33*17^x*17^2*(9^2)^x\) \(=(24^2)^x*(33*17)^x*(24*33)*17^2*(9^2)^x\) considering only the unit digits; \(=(6)^x*(1)^x*2*9*(1)^x\) \(=6*1*2*9*1\) \(=8\)
Answer (D)
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Re: If x is a positive integer, what is the units digit of [#permalink]
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13 Jul 2012, 09:40
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solved in 00:34
consider x =1 24^3 * 33^2 * 17^3 * 9^2
units digit will be 4*9*3*1
=> 8
satisfies for any positive integer.



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Re: If x is a positive integer, what is the units digit of [#permalink]
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11 Sep 2013, 00:32
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Let value of x;
x=1
24^3*33^2*17^3*9^2
unit digits = 4*9*3*1 = 108
unit digit = 8
answer = D



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Re: If x is a positive integer, what is the units digit of [#permalink]
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13 Jun 2014, 20:39
Bunuel wrote: Merging similar topics. Hi Bunuel, In these kind of questions where in it is asked that X is a positive integer, is substituting and value of X a good idea to solve it quickly. Though by taking X=1 or 2 i have arrived at unit's digit as 8 but will it hold for all values of X. Thanks



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Re: If x is a positive integer, what is the units digit of [#permalink]
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14 Jun 2014, 01:33



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Re: If x is a positive integer, what is the units digit of [#permalink]
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14 Jun 2014, 06:24
24^2x+1 for this 2x+1 is odd therefore the unit digit is 4. 33^x+1 can be clubbed with 17^x+2 which means 17^(x+2) can be written as 17^(x+1)*17. So we can write (33*17)^(x+1)*17. 33*17 gives unit digit as 1. Therefore we can write 1^(x+1)*17 = unit digit as 7. 9^2x gives unit digit as 1 since 9 is raised to an even no. of power. so, the total equation becomes 4*1*7*1 = unit digit as 8.



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Re: If x is a positive integer, what is the units digit of [#permalink]
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05 Jun 2015, 01:35
Bunuel wrote: snehamd1309 wrote: Bunuel wrote: Merging similar topics. Hi Bunuel, In these kind of questions where in it is asked that X is a positive integer, is substituting and value of X a good idea to solve it quickly. Though by taking X=1 or 2 i have arrived at unit's digit as 8 but will it hold for all values of X. Thanks Yes. There is only one correct answer in a PS question, thus every x should give the same correct answer. Units digits, exponents, remainders problems directory: newunitsdigitsexponentsremaindersproblems168569.htmlHope it helps. Is my logic right ( i tried to plug some numbers and get to validate my logic. But i am not sure if this logic can be generalized) (24)^(2x + 1) (33)^(x + 1) (17)^(x + 2) (9)^(2x) this can be written in terms of unit digits as : (4)^(2x + 1) (3)^(x + 1) (7)^(x + 2) (9)^(2x) Then (4)^(2x + 1) : give unit digit 4 (3)^(x + 1) : gives unit digit based 3^x and 3 (7)^(x + 2) : give unit digit based 7^x+2 and 7^2 > 7^2 has unit digit 9 (9)^(2x) : give unit digit 1 therefore the terms can be reduced to in unit digits) 4* 3* 3^x * 7^x * 9 * 1 = 2*9 * 3^x *7^x = 8 * 3^x *7^x = 8 * (21)^x ( can i combine the two unit digits 3^ x * 7^ x = (21)^x = 1^x)= 8 * 1^ x = unit digit = 8 my only doubt can i generalize this logic : 3^ x * 7^ x = (21)^x = 1^x to all unit integers.



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Re: If x is a positive integer, what is the units digit of [#permalink]
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05 Jun 2015, 04:41
Jam2014 wrote: Bunuel wrote: snehamd1309 wrote: Merging similar topics. Hi Bunuel, In these kind of questions where in it is asked that X is a positive integer, is substituting and value of X a good idea to solve it quickly. Though by taking X=1 or 2 i have arrived at unit's digit as 8 but will it hold for all values of X. Thanks Yes. There is only one correct answer in a PS question, thus every x should give the same correct answer. Units digits, exponents, remainders problems directory: newunitsdigitsexponentsremaindersproblems168569.htmlHope it helps. Is my logic right ( i tried to plug some numbers and get to validate my logic. But i am not sure if this logic can be generalized) (24)^(2x + 1) (33)^(x + 1) (17)^(x + 2) (9)^(2x) this can be written in terms of unit digits as : (4)^(2x + 1) (3)^(x + 1) (7)^(x + 2) (9)^(2x) Then (4)^(2x + 1) : give unit digit 4 (3)^(x + 1) : gives unit digit based 3^x and 3 (7)^(x + 2) : give unit digit based 7^x+2 and 7^2 > 7^2 has unit digit 9 (9)^(2x) : give unit digit 1 therefore the terms can be reduced to in unit digits) 4* 3* 3^x * 7^x * 9 * 1 = 2*9 * 3^x *7^x = 8 * 3^x *7^x = 8 * (21)^x ( can i combine the two unit digits 3^ x * 7^ x = (21)^x = 1^x)= 8 * 1^ x = unit digit = 8 my only doubt can i generalize this logic : 3^ x * 7^ x = (21)^x = 1^x to all unit integers.Hi jam, Yes, We can generalize this principle for all the Integers with more than 1 digit only Reason: In calculation of the Unit Digit, only Unit Digit matters and all the digits other than unit digit of numbers become redundant.i.e. \((857)^x\) will have same Unit Digit as \(7^x\)I hope clears the doubts!!!
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Re: If x is a positive integer, what is the units digit of [#permalink]
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05 Jun 2015, 04:54
Quote: If x is a positive integer, what is the units digit of (24)^(2x + 1)*(33)^(x + 1)*(17)^(x + 2)*(9)^(2x)?
(A) 4 (B) 6 (C) 7 (D) 8 (E) 9 Here is another method to answer this question very quicklyJust observe the Language of the question " If x is a positive integer, what is the units digit of (24)^(2x + 1)*(33)^(x + 1)*(17)^(x + 2)*(9)^(2x)?" The "is" part confirms that the result of this question will be unique for any value of x which is a positive Integer. Hence this question becomes much easier for any chosen positive integer value of x, Let's take x = 1Now the question becomes (24)^(2x + 1)*(33)^(x + 1)*(17)^(x + 2)*(9)^(2x) = (24)^(2 + 1)*(33)^(1 + 1)*(17)^(1 + 2)*(9)^(2) IMPORTANT POINT : In calculation of the Unit Digit, only Unit Digit matters and all the digits other than unit digit of numbers become redundant.But (24)^(2 + 1) will have same unit digit as 4^(2+1) i.e. 4^3 i.e. 4and But (33)^(1 + 1) will have same unit digit as 3^(1+1) i.e. 3^2 i.e. 9and But (17)^(1 + 2) will have same unit digit as 7^(1+2) i.e. 7^3 i.e. 3and But (9)^(2) will be 1i.e. Unit digit of (24)^(2x + 1)*(33)^(x + 1)*(17)^(x + 2)*(9)^(2x) = 4 x 9 x 3 x 1 = 8 Answer: Option
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Re: If x is a positive integer, what is the units digit of [#permalink]
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17 Mar 2016, 00:00
If x is a positive integer, what is the units digit of (24)^(2x + 1)*(33)^(x + 1)*(17)^(x + 2)*(9)^(2x)?
((24^2)*(33)*(17)*(9^2))^x * (24*33*17^2) Considering only unit digits (6*3*7*1)^x * (4*3*9)
Again reducing to unit digits 6^x * 8 8
Hence D.
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Re: If x is a positive integer, what is the units digit of [#permalink]
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18 Mar 2016, 02:17
(24)^(2x + 1)*(33)^(x + 1)*(17)^(x + 2)*(9)^(2x)=?
2x+1=odd, so 4^odd=4 as unit
2x=even, so 9^even=1 as unit
x+1 and x+2 means that exponent of 7 is one more than exponent of 3. If we look cyclicity we always get 7 in unit when multiplying
So, 4*1*7=8 as unit
D



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Re: If x is a positive integer, what is the units digit of [#permalink]
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29 May 2016, 08:27
kevincan wrote: Remember that in the GMAT, it pays to look for shortcuts!
33^(x+1)*17^(x+2)= ((33*17)^(x+1))*17 which has a units digit of 7! I don't understand the shortcut. Could you kindly enunciate?



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Re: If x is a positive integer, what is the units digit of [#permalink]
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31 May 2016, 13:41
nishi999 wrote: kevincan wrote: Remember that in the GMAT, it pays to look for shortcuts!
33^(x+1)*17^(x+2)= ((33*17)^(x+1))*17 which has a units digit of 7! I don't understand the shortcut. Could you kindly enunciate? Basically, 17^(x+2) = 17^(x+1) * 17 (we take one 17 away from the power to get x+1 instead of x+2 Thus, 33^(x+1) x 17^(x+2) = 33^(x+1) x 17^(x+1) * 17 = (33*17)^(x+1) * 17 We can do the same with (24)^(2x + 1) * (9)^(2x) = (24*9)^(2x) * 24 From (3 3*1 7)^(x+1) * 1 7 we take first two unit digits, first, 3 x 7 = 2 1, then 1 * 7 = 7From (2 4* 9)^(2x) * 2 4 we take first two unit digits, first, 4 * 9 = 3 6, then 6 * 4 = 24 Finally, we have 7 and 2 4 or 7 * 4 = 2 8 Answer: 8 I hope I could help
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Re: If x is a positive integer, what is the units digit of [#permalink]
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28 Aug 2017, 05:30
Hi There is a relatively easier way. As x is a positive number, take x=1. So, 24^3 * 33^2 * 17^3 * 19^2 = 4*9*3*1 = 8. Cheers



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Re: If x is a positive integer, what is the units digit of [#permalink]
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31 Aug 2017, 10:19
MA wrote: If x is a positive integer, what is the units digit of (24)^(2x + 1)*(33)^(x + 1)*(17)^(x + 2)*(9)^(2x)?
A. 4 B. 6 C. 7 D. 8 E. 9 Since we are only concerned with the units digit, we can simplify the expression as: (4)^(2x + 1)*(3)^(x + 1)*(7)^(x + 2)*(9)^(2x) This simplified expression will have the same units digit as the given expression. Next, let’s look at the units digit patterns of powers of 4, 3, 7, and 9, respectively: Units digits of powers of 4: 46 (the patterns repeats in a cycle of 2 with 4^odd = 4 and 4^even = 6) Units digits of powers of 3: 3971 (the patterns repeats in a cycle of 4 with 3^(a multiple of 4) = 1) Units digits of powers of 7: 7931 (the patterns repeats in a cycle of 4 with 7^(a multiple of 4) = 1) Units digits of powers of 9: 91 (the patterns repeats in a cycle of 2 with 9^odd = 9 and 9^even = 1) Since 2x + 1 is always odd regardless of what integer x is, 4^(2x + 1) = 4^odd = 4. Similarly, since 2x is always even regardless of what integer x is, 9^(2x) = 9^even = 1. However, since x + 1 (the exponent of 3) and x + 2 (the exponent of 7) are sometimes odd and sometimes even depending on what integer x is, we are going to change tactics in analyzing the units digit of (3)^(x + 1)*(7)^(x + 2). Notice that: (3)^(x + 1)*(7)^(x + 2) = (3)^(x + 1)*(7)^(x + 1)*7 = (3*7)^(x + 1)*(7) = (21)^(x + 1)*(7) Since we are only concerned with the units digit, we can simplify (21)^(x + 1)*(7) as (1)^(x + 1)*(7). Since 1 raised to any power is 1, the units digit of (1)^(x + 1)*(7) or (21)^(x + 1)*(7) is 1*7 = 7. With this, we can see that the units digit of (4)^(2x + 1)*[(3)^(x + 1)*(7)^(x + 2)]*(9)^(2x) is 4*[7]*1 = 28, i.e., 8. Answer: D
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