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A certain computer program generates 4-digit even numbers using the f  [#permalink]

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GMATbuster's Weekly Quant Quiz#8 Ques #10

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A certain computer program generates 4-digit even numbers using the following numbers: O, 1, 2, 3, 4, and 5. If the digits can be repeated, what Is the probability that a number formed by this program will be a multiple of 4?
A) 16.7%
B) 25%
C) 33.3%
D) 50%
E) 75%

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Re: A certain computer program generates 4-digit even numbers using the f  [#permalink]

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C . Total ways 540. Favourable ways 180, p= 33.35%
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Re: A certain computer program generates 4-digit even numbers using the f  [#permalink]

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Let number formed be X X X X. First three spots can be taken by 6 digits each. Last spot can be taken by 0, 2, 4 only for it to form an even number.
So total possible = 6*6*6*3

For number to be multiple of 4, we need last two digits to be 00,04,12,20,24,32,40,44,52, so 9 possible values.
First two spots can be taken by 6 digits each.
So number of values = 6*6*9

Probability = 6*6*9/6*6*6*3 = 1/2 = 50%

Hence D
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GMAT 1: 610 Q48 V25 GMAT 2: 690 Q50 V32 GMAT 3: 710 Q50 V36 Re: A certain computer program generates 4-digit even numbers using the f  [#permalink]

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A certain computer program generates 4-digit even numbers using the following numbers: O, 1, 2, 3, 4, and S. If the digits can be repeated, what Is the probability that a number formed by this program will be a multiple of 4?

We have
O,1,2,3,4,S

Repetition is allowed

Total no. of cases: 6*6*6*2= 432(we multiply by 2 since the no. of even numbers =2,4)

Total no. of cases where it is divisible by 4:

Cases last 2 digits are div by 4: 24,44 (42 and 22 not eligible)

6*6*2=72

72/432=16.7

Option A Ans
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GMAT 1: 540 Q49 V16 GMAT 2: 680 Q49 V33 A certain computer program generates 4-digit even numbers using the f  [#permalink]

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I would like to go with the solution below :

The question basically asks a us : (numbers div by 4 out of those generated from 0,1,2,3,4 & 5) / (numbers div by 2 out of those generated from 0,1,2,3,4 & 5) .

Say , the numbers div by 2 out of those generated from 0,1,2,3,4 & 5 is x.
Definitely numbers div by 4 out of those generated from 0,1,2,3,4 & 5 is x/2 ( as in every two consecutive even integers , one is div by 4 and the other is not )

Hence , P = (x/2)/x=1/2 =50% Ans D.
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Re: A certain computer program generates 4-digit even numbers using the f  [#permalink]

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even numbers divisible by 4::

--00
--04
--12
--32
--40
--44
--52

=(7*5*6/6*6*6*6)*100= ans A
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A certain computer program generates 4-digit even numbers using the f  [#permalink]

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Total possible number of 4 digit even numbers with the given number = 5*6*6*3 =540
If a number is to be divisible by 4, then the last 2 digits should be divisible by 4.
If the 3rd digit is 0, the possible values for 4th digit are 0 or 4. so, possibilities of a number divisible by 4 and ending in even when 3rd digit is 0 = 5*6*1*2 = 60
If the 3rd digit is 1, the possible value for 4th digit is 2. so, possibilities of a number divisible by 4 and ending in even when 3rd digit is 1 = 5*6*1*1 = 30
If the 3rd digit is 2, the possible values for 4th digit are 0 or 4. so, possibilities of a number divisible by 4 and ending in even when 3rd digit is 2 = 5*6*1*2 = 60
If the 3rd digit is 3, the possible value for 4th digit is 2. so, possibilities of a number divisible by 4 and ending in even when 3rd digit is 3 = 5*6*1*1 = 30
If the 3rd digit is 4, the possible values for 4th digit are 0 or 4. so, possibilities of a number divisible by 4 and ending in even when 3rd digit is 4 = 5*6*1*2 = 60
If the 3rd digit is 5, the possible value for 4th digit is 2. so, possibilities of a number divisible by 4 and ending in even when 3rd digit is 5 = 5*6*1*1 = 30
Total possibilities of a 4 digit number divisible by 4 and ending in even = (60+30+60+30+60+30)=270
probability= 270/540 = 50%

OPTION D

Originally posted by sowmiyav on 10 Nov 2018, 12:11.
Last edited by sowmiyav on 13 Nov 2018, 01:00, edited 1 time in total.
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Re: A certain computer program generates 4-digit even numbers using the f  [#permalink]

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We know that there are in all, total of 6 digits to choose from.

Letter O, 1, 2, 3, 4, and letter S.

1st case: (different numbers that are a multiple and hence divisible by 4)

For any number to be divisible by 4, the rule is that it's last two digits have to be divisible by 4.

_ _ _ _ (4 digit numbers that can be formed)

Using the digits available and knowing that repetition is allowed, we have only two different ways of filling the last two spots -
24
44

Therefore, the number of ways in which we can have the 4 digit number divisible by 4 is

6 6 2 (since first two places are allowed repetition and there are 6 digits to choose from)

6*6*2 ways

2nd case (total number of even 4 digit numbers that can be formed)

From the digits available, the last digit has to either be 2 or 4 for the 4-digit number to be even.

Therefore, number of ways possible -
6*6*6*2 ways

Concluding the solution-

Probability of having a 4-digit number divisible by 4, therefore, can be calculated as below

= (6*6*2 / 6*6*6*2) * 100
= (1/6) * 100
= 16.67%

Ans. (A) 16.67%

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Re: A certain computer program generates 4-digit even numbers using the f  [#permalink]

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Official Solution: Attachment: S10.jpg [ 93.41 KiB | Viewed 1511 times ]

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Re: A certain computer program generates 4-digit even numbers using the f  [#permalink]

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No calculations needed I guess:

Every second even number can be divided by 4.
That means 50%.
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Re: A certain computer program generates 4-digit even numbers using the f  [#permalink]

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MichaelGolth wrote:
No calculations needed I guess:

Every second even number can be divided by 4.
That means 50%.

Hello...

very interesting......can you elaborate on this please?

regards
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Re: A certain computer program generates 4-digit even numbers using the f  [#permalink]

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MichaelGolth, Mansoor50

"MichaelGolth wrote:
No calculations needed I guess:

Every second even number can be divided by 4.
That means 50%."

Above approach is valid only when number of even numbers are even.

For example, suppose set of even numbers are (2,4,6). Here no of multiple of 4 is not half the total no of even numbers.

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Re: A certain computer program generates 4-digit even numbers using the f  [#permalink]

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Hi gladi,just a prompt question for you, based on a divisibility rule (it is also written in the OA) if the last 2 digits of a number are divisible by 4 then the whole number is divisible by 4 right? However I was wondering why in this case the choices which end in 00 are also included is it also part of a rule or just an assumption?
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Re: A certain computer program generates 4-digit even numbers using the f  [#permalink]

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UNSTOPPABLE12 wrote:

Hi gladi,just a prompt question for you, based on a divisibility rule (it is also written in the OA) if the last 2 digits of a number are divisible by 4 then the whole number is divisible by 4 right? However I was wondering why in this case the choices which end in 00 are also included is it also part of a rule or just an assumption?

Because 00 or simply 0 is also divisible by 4. Recall that 0 is divisible by every integer except 0 itself.
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GMAT 1: 740 Q50 V40 GMAT 2: 770 Q51 V42 Re: A certain computer program generates 4-digit even numbers using the f  [#permalink]

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Hi UNSTOPPABLE12, Hope you are doing great! Please find my responses in red to your specific questions -
UNSTOPPABLE12 wrote:

Hi gladi,just a prompt question for you, based on a divisibility rule (it is also written in the OA) if the last 2 digits of a number are divisible by 4 then the whole number is divisible by 4 right? Perfect - this is the divisibility test of 4 However I was wondering why in this case the choices which end in 00 are also included is it also part of a rule or just an assumption? 00 is also divisible by 4 - 0 in fact is divisible by all integers

Also, have you checked out the more elegant solution by u1983? Once we confirm that an even number of total even 4-digit numbers are present it becomes obvious that half of them will be divisible by 4.
Why am I putting this additional rule? Let's say we have 2,4,6 ... only 4 is divisible by 4. But if we have 2,4,6,8 - 2/4 or 50% are divisible by 4. As long as the total even numbers are even, and consecutive ( which is exactly the case here) we can directly say 50% without computing the actual number of even diivisble by 4.

Even just the method shown in OE is sufficient to get the question in under 2 mins.

Hope it helps.
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Re: A certain computer program generates 4-digit even numbers using the f  [#permalink]

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Thank you Bunuel and Gladiator59 really helpful remarks.
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A certain computer program generates 4-digit even numbers using the f  [#permalink]

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UNSTOPPABLE12 wrote:

Hi gladi,just a prompt question for you, based on a divisibility rule (it is also written in the OA) if the last 2 digits of a number are divisible by 4 then the whole number is divisible by 4 right? However I was wondering why in this case the choices which end in 00 are also included is it also part of a rule or just an assumption?

the number ends with 00 always divisible by 4.
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A certain computer program generates 4-digit even numbers using the f  [#permalink]

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nkin wrote:
Let number formed be X X X X. First three spots can be taken by 6 digits each. Last spot can be taken by 0, 2, 4 only for it to form an even number.
So total possible = 6*6*6*3

For number to be multiple of 4, we need last two digits to be 00,04,12,20,24,32,40,44,52, so 9 possible values.
First two spots can be taken by 6 digits each.
So number of values = 6*6*9

Probability = 6*6*9/6*6*6*3 = 1/2 = 50%

Hence D

The 1st spot can't be taken by '0' coz its clearly mentioned that the digit is '4-digit' => 0441, 0532 these are 3-digits ;
So, total such no.s with repetitions allowed = 5x6x6x3

Similarly to be multiple of 4 the last 2-digits can be 00,04,12,20,24,32,40,44,52. As you mentioned ;
First spot can be taken here as well by '5-digits' and not '6' [0 doesn't counts]
So, total such no.s = 5x6x9

P (divisible by 4) = [5x6x6x3] / [5x6x9] = 1/2 or 50% [NOTE- The answer will be same as in both cases '6' or '5' gets cancelled. ]
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Re: A certain computer program generates 4-digit even numbers using the f  [#permalink]

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mishugarg wrote:
even numbers divisible by 4::

--00
--04
--12
--32
--40
--44
--52

=(7*5*6/6*6*6*6)*100= ans A

What about -20? It’s also divisible by 4.

Posted from my mobile device Re: A certain computer program generates 4-digit even numbers using the f   [#permalink] 22 Jan 2019, 00:00
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