Will Rearrangement Apply here?

Number Theory Level pending

How many 5-digit positive integers can be formed using all the digits 5, 6, 7, 8, 9 such that it is divisible by 21?


The answer is 0.

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1 solution

Nashita Rahman
Jan 28, 2017

For a number to be divisible by 21 , it has to be divisible by both 3 and 7 . For a number to be divisible by 3 , the sum of its digits must be divisible by 3.

In this case , Sum of the digits = 5+6+7+8+9 = 35 which is not divisible by 3. Hence any 5-digit number using 5,6,7,8,9 as its digits will not be divisible by 3. As it cannot be divisible by 3 therefore it cannot be divisible by 21. So the answer is 0.

Nice forehand observation.

Just out of curiosity, suppose we had five digits that together gave a sum divisible by 3, then what will you do to check how many permutations are divisible by 7?

If you're able to find the approach, solve this problem

How many 5-digit positive numbers can be formed by the digits 0, 2, 3, 5 and 8 without repetition of digits such that they are divisible by 21?

Tapas Mazumdar - 4 years, 4 months ago

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Thanks :) The link to the problem that you've given is not opening!

Nashita Rahman - 4 years, 4 months ago

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Actually there's no such problem posted by anybody. It's made by me as a follow-up problem for you.

Tapas Mazumdar - 4 years, 4 months ago

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@Tapas Mazumdar That's nice , I am trying....

Nashita Rahman - 4 years, 4 months ago

@Tapas Mazumdar Okay , sorry then!

Nashita Rahman - 4 years, 4 months ago

Its becoming lengthy ,do you have an easier approach? If yes then let me know!

Nashita Rahman - 4 years, 4 months ago

Tapas Da, can u do it?

Md Zuhair - 4 years, 4 months ago

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@Nashita Rahman , @Md Zuhair

There are a good amount of such numbers.

Some examples are:

20853 28350 25830 23058 20832 20538 32508 35280 38052 58023 53802 80325 82530 85302 20853 \\ 28350 \\ 25830 \\ 23058 \\ 20832 \\ 20538 \\ 32508 \\ 35280 \\ 38052 \\ 58023 \\ 53802 \\ 80325 \\ 82530 \\ 85302

My approach here was to find the remainders left when a certain place value of the given numbers are divided by 7 7 and then use modular arithmetic to devise like this:

( m o d 7 ) { ϕ , 1 , 5 , 6 , 4 } 1 0 4 for { 0 , 2 , 3 , 5 , 8 } respectively { 0 , 5 , 4 , 2 , 6 } 1 0 3 for { 0 , 2 , 3 , 5 , 8 } respectively { 0 , 4 , 6 , 3 , 2 } 1 0 2 for { 0 , 2 , 3 , 5 , 8 } respectively { 0 , 6 , 2 , 1 , 3 } 1 0 1 for { 0 , 2 , 3 , 5 , 8 } respectively { 0 , 2 , 3 , 5 , 1 } 1 0 0 for { 0 , 2 , 3 , 5 , 8 } respectively \begin{aligned} \pmod{7} \equiv & \{\phi , 1 , 5 , 6 , 4 \} \to 10^4 \text{ for } \{ 0 , 2,3,5,8 \} \text{ respectively } \\ & \{0,5,4,2,6\} \to 10^3 \text{ for } \{ 0 , 2,3,5,8 \} \text{ respectively } \\ & \{0,4,6,3,2\} \to 10^2 \text{ for } \{ 0 , 2,3,5,8 \} \text{ respectively } \\ & \{0,6,2,1,3\} \to 10^1 \text{ for } \{ 0 , 2,3,5,8 \} \text{ respectively } \\ & \{0,2,3,5,1\} \to 10^0 \text{ for } \{ 0 , 2,3,5,8 \} \text{ respectively } \end{aligned}

The ϕ \phi above denotes that 0 0 cannot occupy the 1 0 4 10^4 place.

Notice that to form numbers divisible by 7 7 we have to pick out respective numbers from each set such that no two numbers correspond to the same position in the sets (as no repetition is allowed) and such that the sum of the numbers turns out to be a multiple of 7 7 . Since the maximum sum that we can make here is 6 + 6 + 6 + 6 + 0 = 24 6+6+6+6+0 = 24 (without any correspondence), therefore, we have to pick up the numbers that will give 7 , 14 7,14 or 21 21 .

The above mentioned are some of the numbers which I was able to find by some observation. If there's any further shortcut we can apply to find these numbers, please let me know.

Tapas Mazumdar - 4 years, 4 months ago

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@Tapas Mazumdar There may be a shortcut to find the numbers which I don't know but your approach is also good. Thanks!

Nashita Rahman - 4 years, 4 months ago

Tapas Da, Is there only 1 number? That is what I am getting,

28350

Md Zuhair - 4 years, 4 months ago

@Nashita Rahman @Tapas Mazumdar @Md Zuhair I have approached like this Sum of the digits is divisible by 3 so every number formed by this will be divisible by three.Therefore I am just concentrating on the divisibility by 7 7

N = a b c d e N=\overline{abcde} where each digit is from 0 , 2 , 3 , 5 , 8 0,2,3,5,8

Now , N = 10000 a + 1000 b + 100 c + 10 d + e 4 a + 6 b + 2 c + 3 d + e ( m o d 7 ) N=10000 a+ 1000b+100c+10d+e \equiv 4a+6b+2c+3d+e \pmod{7} .As 0 + 2 + 3 + 5 + 8 = 18 4 ( m o d 7 ) 0+2+3+5+8=18 \equiv 4 \pmod{7}

7 4 a + 6 b + 2 c + 3 d + e 7 3 a + 5 b + c + 2 d + ( a + b + c + d + e ) 7 3 a + 5 b + c + 2 d + 4 7 2 a + 4 b + d + ( a + b + c + d + 4 ) 7 2 a + 4 b + d + 18 + 4 e 7 2 a + 4 b + d e + 1 7 | 4a+6b+2c+3d+e \\ 7| 3a+5b+c+2d +(a+b+c+d+e) \\ 7 | 3a+5b+c+2d +4 \\ 7 | 2a+4b+d +(a+b+c+d+4) \\ 7| 2a+4b+d +18+4-e \\ 7| 2a+4b+d-e+1

Here a 0 a \neq 0 so a = 2 , 3 , 5 , 8 a=2,3,5,8 and varying b , d , e b,d,e we can find such numbers.

P.S. I believe this method is not time -saving there must be some proper method

Kushal Bose - 4 years, 3 months ago

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