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?
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
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?
Log in to reply
Thanks :) The link to the problem that you've given is not opening!
Log in to reply
Actually there's no such problem posted by anybody. It's made by me as a follow-up problem for you.
Log in to reply
@Tapas Mazumdar – That's nice , I am trying....
@Tapas Mazumdar – Okay , sorry then!
Its becoming lengthy ,do you have an easier approach? If yes then let me know!
Tapas Da, can u do it?
Log in to reply
There are a good amount of such numbers.
Some examples are:
2 0 8 5 3 2 8 3 5 0 2 5 8 3 0 2 3 0 5 8 2 0 8 3 2 2 0 5 3 8 3 2 5 0 8 3 5 2 8 0 3 8 0 5 2 5 8 0 2 3 5 3 8 0 2 8 0 3 2 5 8 2 5 3 0 8 5 3 0 2
My approach here was to find the remainders left when a certain place value of the given numbers are divided by 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
The ϕ above denotes that 0 cannot occupy the 1 0 4 place.
Notice that to form numbers divisible by 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 . Since the maximum sum that we can make here is 6 + 6 + 6 + 6 + 0 = 2 4 (without any correspondence), therefore, we have to pick up the numbers that will give 7 , 1 4 or 2 1 .
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.
Log in to reply
@Tapas Mazumdar – There may be a shortcut to find the numbers which I don't know but your approach is also good. Thanks!
Tapas Da, Is there only 1 number? That is what I am getting,
28350
@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
N = a b c d e where each digit is from 0 , 2 , 3 , 5 , 8
Now , N = 1 0 0 0 0 a + 1 0 0 0 b + 1 0 0 c + 1 0 d + e ≡ 4 a + 6 b + 2 c + 3 d + e ( m o d 7 ) .As 0 + 2 + 3 + 5 + 8 = 1 8 ≡ 4 ( m o d 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 + 1 8 + 4 − e 7 ∣ 2 a + 4 b + d − e + 1
Here a = 0 so a = 2 , 3 , 5 , 8 and varying b , d , e we can find such numbers.
P.S. I believe this method is not time -saving there must be some proper method
Problem Loading...
Note Loading...
Set Loading...
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.