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How many four digit numbers that are divisible by 4 can be formed usin

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How many four digit numbers that are divisible by 4 can be formed usin  [#permalink]

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New post Updated on: 27 Oct 2010, 23:20
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How many four digit numbers that are divisible by 4 can be formed using the digits 0 to 7 if no digit is to occur more than once in each number?

A. 520
B. 370
C. 345
D. 432
E. 353

Originally posted by ankitranjan on 27 Oct 2010, 02:49.
Last edited by ankitranjan on 27 Oct 2010, 23:20, edited 1 time in total.
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Re: How many four digit numbers that are divisible by 4 can be formed usin  [#permalink]

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New post 27 Oct 2010, 23:36
Answer is B.370
I have solved it in this way....
A number will be divisible by 4 if the number formed by its last two digit is divisible by 4.
And in this way we have its last two digit as ...
04,12,16,20,24,32,36,40,52,56,60,64,72,76 {Four of these includes 0 and remaining ten are without 0}
( I didnt include 44 as repetition is not allowed and others as digit should be less than 8)

Now consider the case when any of the last two digit is 0. we can get 6*5 = 30 different numbers

so total formed number where any of the last two digit is 0 = 4*30=120

Now consider the case when none of the last two digit is 0 = 5 * 5 =25 (because 1000s place cant be filled by 0 or else it will be a three digit number)

so total formed number when none of the last two digit is 0 = 10 * 25 =250

Hence required answer is 250+120 =170.
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Re: How many four digit numbers that are divisible by 4 can be formed usin  [#permalink]

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New post 07 Jan 2018, 09:54
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ankitranjan wrote:
How many four digit numbers that are divisible by 4 can be formed using the digits 0 to 7 if no digit is to occur more than once in each number?

A. 520
B. 370
C. 345
D. 432
E. 353


There is another way of doing this question. This way is not perfect (might not give exact answer) BUT if options are not very close, then this method can be applied.

So, lets first see how many 4-digit numbers we can form using 8 digits (from 0 to 7) such that no digit is repeated. Thousands place can be filled in 7 ways (we cannot have 0 there). For each of these 7 ways, hundreds place can be filled in 7 ways again (0 can come, but the digit already used for thousands place is excluded). Similarly for each of these, tens place can be filled in 6 ways and ones place in 5 ways. So total such numbers are:

7*7*6*5 = 1470.

Now out of these, we have to find out how many are divisible by 4. So answer will be approximately 1/4th of the total, or Approx = 1/4 * 1470 = 367.5.. nearest option to 367.5 is 370.

Hence B answer

(as i said this might not give us exact answer but close enough so as to mark the right option)
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Re: How many four digit numbers that are divisible by 4 can be formed usin  [#permalink]

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New post 13 Jan 2018, 08:37
ankitranjan wrote:
Answer is B.370
I have solved it in this way....
A number will be divisible by 4 if the number formed by its last two digit is divisible by 4.
And in this way we have its last two digit as ...
04,12,16,20,24,32,36,40,52,56,60,64,72,76 {Four of these includes 0 and remaining ten are without 0}
( I didnt include 44 as repetition is not allowed and others as digit should be less than 8)

Now consider the case when any of the last two digit is 0. we can get 6*5 = 30 different numbers

so total formed number where any of the last two digit is 0 = 4*30=120

Now consider the case when none of the last two digit is 0 = 5 * 5 =25 (because 1000s place cant be filled by 0 or else it will be a three digit number)

so total formed number when none of the last two digit is 0 = 10 * 25 =250

Hence required answer is 250+120 =170.


I have confusion with below statements
so total formed number where any of the last two digit is 0 = 4*30=120
and
so total formed number when none of the last two digit is 0 = 10 * 25 =250

Can someone provide further explanation. Thanks.
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Re: How many four digit numbers that are divisible by 4 can be formed usin  [#permalink]

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New post 13 Jan 2018, 12:10
vs224 wrote:
ankitranjan wrote:
Answer is B.370
I have solved it in this way....
A number will be divisible by 4 if the number formed by its last two digit is divisible by 4.
And in this way we have its last two digit as ...
04,12,16,20,24,32,36,40,52,56,60,64,72,76 {Four of these includes 0 and remaining ten are without 0}
( I didnt include 44 as repetition is not allowed and others as digit should be less than 8)

Now consider the case when any of the last two digit is 0. we can get 6*5 = 30 different numbers

so total formed number where any of the last two digit is 0 = 4*30=120

Now consider the case when none of the last two digit is 0 = 5 * 5 =25 (because 1000s place cant be filled by 0 or else it will be a three digit number)

so total formed number when none of the last two digit is 0 = 10 * 25 =250

Hence required answer is 250+120 =170.


I have confusion with below statements
so total formed number where any of the last two digit is 0 = 4*30=120
and
so total formed number when none of the last two digit is 0 = 10 * 25 =250

Can someone provide further explanation. Thanks.


Hi vs224

As you know for divisibility by 4 last two digits should be divisible by 4 and in the above solution by ankitranjan, we know how many last two digits are possible from number between 0 to 7
So first he/she has taken last two digits containing 0 and there are 4 such numbers. now out of 8 number we have already used 2 numbers, hence the hundreds and thousands digits can be arranged in 6*5=30 ways
so total number with last two digits as 0=4*30=120

now for the second part where last two digits are not 0, we have 10 such numbers that are divisible by 4. Thousands digit cannot be 0, so it can take 5 possible numbers out of the remaining numbers and similarly hundreds digit can then take 5 left over numbers. Hence total number = 10*5*5=250
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Re: How many four digit numbers that are divisible by 4 can be formed usin &nbs [#permalink] 13 Jan 2018, 12:10
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