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

Question

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

OptimusPrepJanielle · Answer

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

OptimusPrepJanielle · Answer

Thanks a lot for pointing the typo. Edited my response.

Abhishek009 · Answer

Great way OptimusPrepJanielle

Here is something parallel to OptimusPrepJanielle's way

Attachment:

Units Digit.PNG [ 19.8 KiB | Viewed 6849 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...

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

Alex75PAris · Answer

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

OptimusPrepJanielle · Answer

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

Abhishek009 · Answer

Certainly , it was a typo.... Thanks for pointing out !! :-D

vitaliyGMAT · Answer

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

LeonidK · Answer

great explanation

thanks for sharing

matthewsmith_89 · Answer

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

bumpbot · Answer

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

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

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