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# If x is a positive integer, what is the units digit of

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VP
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If x is a positive integer, what is the units digit of [#permalink]

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11 Jul 2006, 09:32
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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
[Reveal] Spoiler: OA

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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, 10:26
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, 10:44
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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, 13:14
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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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11 Jul 2006, 18:39
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.

Last edited by MA on 11 Jul 2006, 19:31, 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, 22:45
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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, 08:40
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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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10 Sep 2013, 23:32
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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, 19:39
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, 00:33
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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: new-units-digits-exponents-remainders-problems-168569.html

Hope it helps.
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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, 05:24
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, 00:35
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: new-units-digits-exponents-remainders-problems-168569.html

Hope 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, 03:41
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Expert's post
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: new-units-digits-exponents-remainders-problems-168569.html

Hope 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, 03:54
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 quickly

Just 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 = 1

Now 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. 4

and But (33)^(1 + 1) will have same unit digit as 3^(1+1) i.e. 3^2 i.e. 9

and But (17)^(1 + 2) will have same unit digit as 7^(1+2) i.e. 7^3 i.e. 3

and But (9)^(2) will be 1

i.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
[Reveal] Spoiler:
D

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Re: If x is a positive integer, what is the units digit of [#permalink]

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16 Mar 2016, 23:00
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.

Thanks

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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, 01:17
(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, 07:27
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, 12:41
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 (33*17)^(x+1) * 17 we take first two unit digits, first, 3 x 7 = 21, then 1 * 7 = 7
From (24*9)^(2x) * 24 we take first two unit digits, first, 4 * 9 = 36, then 6 * 4 = 24
Finally, we have 7 and 24 or 7 * 4 = 28

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, 04:30
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, 09:19
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: 4-6 (the patterns repeats in a cycle of 2 with 4^odd = 4 and 4^even = 6)

Units digits of powers of 3: 3-9-7-1 (the patterns repeats in a cycle of 4 with 3^(a multiple of 4) = 1)

Units digits of powers of 7: 7-9-3-1 (the patterns repeats in a cycle of 4 with 7^(a multiple of 4) = 1)

Units digits of powers of 9: 9-1 (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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Re: If x is a positive integer, what is the units digit of   [#permalink] 31 Aug 2017, 09:19
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