Question

What 3 digits should we add to number 523... (instead of dots), so that it would be divisible by 7, 8, 9?

Answer 1

It really seemed to be a question for Chinese Remainder theorem, however, I don't think I can apply it here...

523000 + 100x + 10y + z.
Using divisibility rules I could make a few equations:
x + y + z = 8k
523000 + 100 x + 10y - 2z = 7n
10 + x + y + z = 9p

Simplifying them a little bit I get:
x + y + z = 8k
2 + 2x + 3y - 2z = 7n
1 + x + y + z = 9p

However I have no idea what to do next...

Answer 2

(oh, just n and p after simplification should be different numbers, sorry for my mistake!)

Answer 3

We should be able to do this simply by using divisibility rules

Answer 4

An integer is divisible by 9 if the sum of its digits are divisible by 9

Answer 5

Well, yes, that's what I wrote with my equations:
x + y + z = 8k //divisibility by 8
523000 + 100 x + 10y - 2z = 7n //divisibility by 7
10 + x + y + z = 9p //divisibility by 9

Answer 6

Ahh right, your equations represent the same..

Answer 7

stare at below two equations

```
x + y + z = 8k
1 + x + y + z = 9p
```

Is it easy to see that x+y+z has to equal 8 ?

Answer 8

Keep in mind that x, y, z are digits between 0 and 9

Answer 9

Oh, I think that I had messed up with x + y + z = 8k... It should have been 100x + 10y + z = 8k.

Answer 10

Oh riht, I'm also not fully awake haha

Answer 11

So, when simplifying it would get to 4x + 2y + z = 8n

Answer 12

Take 7*8*9 = 504
and find a number between 523000 and 523999
that is a multiple of 504
use excel or brute force

Answer 13

@retirEEd the problem is that I am only allowed to use my head. ;)

Answer 14

That wasn't in the statement of the problem, so sorry I don't accept the response.

Answer 15

Will the solution be always unique??

Answer 16

@Zyberg - I think the solution given by @retirEEd is the fastest way to solve this problem. No excel is needed however..

Answer 17

Start by finding 523000 mod 504

Answer 18

352

Answer 19

there will be 2 solutions then

Answer 20

That means 523000 + 352 is divisible by 504

Answer 21

Yeah looks two solutions

Answer 22

ganeshie, shouldn't that be 523000 + 152?

Answer 23

But dang, I see that it's very simple! So weird that I tried to solve it by using divisibility rules and such ;)
Thank you as always, for pointing the right way to me! :)

Answer 24

Thank @retirEEd and please medal him he really helped us :)

Answer 25

Hey why are you adding 152 ?

Answer 26

I fanned him... That method is the easiest

Answer 27

Well, if 523000 is congruent to 352 in mod 504, that means that 523000 - 352 is divisible by 504. However, it is out of bounds, so we add 504 to it and we get -352+504 =152 ;)

Answer 28

sure, after getting 523152 we would need to add 504 to the number to get the next one divisible by 504.

Answer 29

Wow I was so distracted. Thank you for explaining :)

Answer 30

The three numbers to add are 152 and I didn't use excel but it would have been easier then the using a simple calculator.