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If integers ending in zero are excluded, what is the units digit of th

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Bunuel
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If integers ending in zero are excluded, what is the units digit of th [#permalink] New post   27 Apr 2016, 23:33
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If integers ending in zero are excluded, what is the units digit of the product of the even integers from 202 to 298, inclusive?

A. 2
B. 4
C. 6
D. 8
E. 9
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C
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OptimusPrepJanielle
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If integers ending in zero are excluded, what is the units digit of th [#permalink] New post   Updated on: 03 May 2016, 23:58
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Bunuel wrote:
If integers ending in zero are excluded, what is the units digit of the product of the even integers from 202 to 298, inclusive?

A. 2
B. 4
C. 6
D. 8
E. 9


In such questions, it is of utmost importance that you select the list in a smart way.
Following will be the list of even integers:

202, 204, 206, 208 - Units digit of the product = units digit of 2*4*6*8 = 4
212, 214, 216, 218
222, 224, 226, 228
232, 234, 236, 238
.
.
.
.
282, 284, 286, 288
292, 294, 296, 298

We have 10 such sets, the units digit of which is 4
Hence the units digit of the whole product = units digit of 4^10
Since the cyclicity of 4 is 2, units digit of 4^10 = units digit of 4^2 = 6

Correct Option: C

Originally posted by OptimusPrepJanielle on 28 Apr 2016, 00:33.
Last edited by OptimusPrepJanielle on 03 May 2016, 23:58, edited 1 time in total.
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OptimusPrepJanielle
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Re: If integers ending in zero are excluded, what is the units digit of th [#permalink] New post   03 May 2016, 23:59
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trkswr wrote:

Cyclicity of 4 is 2


Thanks a lot for pointing the typo. Edited my response.
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Re: If integers ending in zero are excluded, what is the units digit of th [#permalink] New post   04 May 2016, 12:49
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Great way OptimusPrepJanielle

Here is something parallel to OptimusPrepJanielle's way

Attachment:
Units Digit.PNG
Units Digit.PNG [ 19.8 KiB | Viewed 4570 times ]


Jeneli is taking product of each row as 4 to arrive at the product of units digit.

However I feel it can also be done simply by taking columns ( As there will be only 4 columns )

Posting the approach , plz feel free to revert in case of any question , both OptimusPrepJanielle & I will be more than happy to discuss further on this issue... :-D


PS : KUDOS OptimusPrepJanielle for the excellent approach, it will be better if we can further workout any more shortcuts, lets do some brainstorming...
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Alex75PAris
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Re: If integers ending in zero are excluded, what is the units digit of th [#permalink] New post   04 May 2016, 13:10
202 x 204 x 206 x 208 => unit digit 4
212 x 214 x 216 x 218 => unit digit 4
...
292 x 294 x 296 x 298 => unit digit 4

So we have 10 numbers ending by 4 to multiply.

Given that 4 x 4 = 16, we have 5 numbers ending by 6 to multiply
Given that 6 x 6 = 36, we have 3 numbers ending by 6 to multiply

So the unit digit is 6

Answer choice C
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OptimusPrepJanielle
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Re: If integers ending in zero are excluded, what is the units digit of th [#permalink] New post   05 May 2016, 02:51
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Abhishek009 wrote:
Great way OptimusPrepJanielle

Here is something parallel to OptimusPrepJanielle's way

Attachment:
Units Digit.PNG


Jeneli is taking product of each row as 4 to arrive at the product of units digit.

However I feel it can also be done simply by taking columns ( As there will be only 4 columns )

Posting the approach , plz feel free to revert in case of any question , both OptimusPrepJanielle & I will be more than happy to discuss further on this issue... :-D


PS : KUDOS OptimusPrepJanielle for the excellent approach, it will be better if we can further workout any more shortcuts, lets do some brainstorming...


I must say, that is a very detailed comparison :D
Will surely post short cuts for problems as and when I find them.

P.S. You might want to edit the typo in column 4. It should be 8^10 :)
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Re: If integers ending in zero are excluded, what is the units digit of th [#permalink] New post   05 May 2016, 07:58
OptimusPrepJanielle wrote:
Abhishek009 wrote:
Great way OptimusPrepJanielle

Here is something parallel to OptimusPrepJanielle's way

Attachment:
Units Digit.PNG


Jeneli is taking product of each row as 4 to arrive at the product of units digit.

However I feel it can also be done simply by taking columns ( As there will be only 4 columns )

Posting the approach , plz feel free to revert in case of any question , both OptimusPrepJanielle & I will be more than happy to discuss further on this issue... :-D


PS : KUDOS OptimusPrepJanielle for the excellent approach, it will be better if we can further workout any more shortcuts, lets do some brainstorming...


I must say, that is a very detailed comparison :D
Will surely post short cuts for problems as and when I find them.

P.S. You might want to edit the typo in column 4. It should be 8^10 :)


Certainly , it was a typo....

Thanks for pointing out !! :-D :P
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Re: If integers ending in zero are excluded, what is the units digit of th [#permalink] New post   11 Nov 2016, 04:23
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We start from \(202*204*206*208*\) … and end with ... \(*292*294*296*298\).

The pattern of units digits \(2, 4, 6, 8\) repeats \(10\) times. So we have

\(2^{10}*4^{10}*6^{10}*8^{10}\)

\(6\) will always end in \(6; 4\) to the even power will end in \(6\). \(2\) and \(8\) have cyclicity of \(4\).

\(2^2*6*6*8^2 = 4*6*6*4 = 6\)
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Re: If integers ending in zero are excluded, what is the units digit of th [#permalink] New post   08 Apr 2017, 08:02
OptimusPrepJanielle wrote:
Bunuel wrote:
If integers ending in zero are excluded, what is the units digit of the product of the even integers from 202 to 298, inclusive?

A. 2
B. 4
C. 6
D. 8
E. 9


In such questions, it is of utmost importance that you select the list in a smart way.
Following will be the list of even integers:

202, 204, 206, 208 - Units digit of the product = units digit of 2*4*6*8 = 4
212, 214, 216, 218
222, 224, 226, 228
232, 234, 236, 238
.
.
.
.
282, 284, 286, 288
292, 294, 296, 298

We have 10 such sets, the units digit of which is 4
Hence the units digit of the whole product = units digit of 4^10
Since the cyclicity of 4 is 2, units digit of 4^10 = units digit of 4^2 = 6

Correct Option: C


great explanation :)
thanks for sharing :)
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matthewsmith_89
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Re: If integers ending in zero are excluded, what is the units digit of th [#permalink] New post   08 Apr 2017, 13:38
When the second digit is 0 and all even units (2 - 8) are multiplied, the result is 4
When the second digit is 1 and you carry over the 4 and multiply all even units (2- 8), the result is 6
When the second digit is 2, and you carry over the 6 and multiply all even units (2-8), the result is 4
There is no need to do any more calculations. You can see pattern as you can see on the table attached
The answer is 6 which is answer C
Attachments

Matthew's Explanation.xlsx [9.84 KiB]
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Re: If integers ending in zero are excluded, what is the units digit of th [#permalink] New post   23 Oct 2023, 14:48
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