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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.

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

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11 Jul 2006, 19:39

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, 20:31, edited 2 times in total.

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.

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.

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.

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.

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 digits3^ 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.

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.

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 digits3^ 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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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 (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
_________________

Please kindly +Kudos if my posts or questions help you!

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:

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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