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How to find number of 4 digit numbers divisible by 4 that can be obtained using 1,2,3,4,5,6 with no numbers repeating?

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3 Answers 3

up vote 7 down vote accepted

For non-repeating: number is divisible by 4 iff its last two digits are divisible by 4. IN your case, it will be 12 16 24 32 36 52 56 64

So you have 8 possible endings.

First two digits can be chosen in 4 * 3 ways, so overall answer will be 12 * 8 = 96

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The first two digits can only be chosen in $4\cdot3=12$ ways, so the overall answer is $12\cdot8=96$. –  joriki Aug 18 '11 at 17:28
Thanks, edited my post. –  Oleksandr Kuvshynov Aug 18 '11 at 17:30

[This is an answer to the original question, in which repetition was not excluded.]

For the number to be divisible by $4$, the number formed by the last two digits must be divisible by $4$. With the digits $1$ through $6$, you can use the $3$ last digits $2$, $4$ and $6$, and each of these can be combined with half of the digits to form a $2$-digit number divisible by $4$, so there are $9$ combinations for the last two digits. The other two digits can be chosen arbitrarily, so there are $36$ combinations for those, for a total of $9\cdot 36=324$ different numbers.

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I forgot to say that none of the nos should be repeated... –  S.M.09 Aug 18 '11 at 17:24
Then having your $9$ possibilities for the last two digits, you can choose $4$ numbers for the first digit and $3$ for the second, giving $9\cdot 4 \cdot 3=108$ –  Ross Millikan Aug 18 '11 at 17:27
@Ross: There's only $8$ for the last two because one of the $9$ is $44$ with repetition. –  joriki Aug 18 '11 at 17:29
@joriki: right you are. So $96$ total. –  Ross Millikan Aug 18 '11 at 18:16

Explanation: Take 5 blanks _ _ _ _ _. Since the number has to be divisible by 4 we can have only the following cases 12, 16, 24, 32, 36, 52, 56, 64.That is you can fill the last two blanks in eight ways. Now take the first three blanks. Since we already selected 2 numbers, we are left with 4 numbers. That is we can fill the first blank in 4 ways, similarly the second in 3, the third in 2 ways. By fundamental principle of counting we have to multiply all of them. That is 4 x 3 x 2 x 8=192.

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