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Dsds7292 wrote:
Find a number consisting of 9 digits in which each of the digits from 1 to 9 appears only once. This number should satisfy the following requirements:
a. The number should be divisible by 9.
b. If the most right digit is removed, the remaining number should be divisible by 8.
c. If then again the most right digit is removed, the remaining number should be divisible by 7.
d. etc. until the last remaining number of one digit which should be divisible by 1.
thaithai wrote:
such as 8 is divisible by 432 => 8 is divisible by 8432.
the number is ?????????
If the most right digit is removed, the remaining number should be divisible by 8 => 8????????
If then again the most right digit is removed, the remaining number should be divisible by 7 => 87???????
If then again the most right digit is removed, the remaining number should be divisible by 6 => 876??????
If then again the most right digit is removed, the remaining number should be divisible by 5 => 876?????5
so on. 876?4???5
so on. 876?43??5.
so on. 876?432?5. now due to: 3 is divisible by 215. 3 is not divisible 295 => 876943215 or 876943125
If then again the most right digit is removed, the remaining number should be divisible by 2 => 876943125 .we remove 5 digit to 87694312 divide 2
result is 876943125
Dsds7292 wrote:
I am sorry, but 8769431 /7 =1252775.8571428571428571428571429, which isn't divisible.
thaithai wrote:
do u mean " most right digit is remove"= "remove last right digit"
or " most right digit is remove"= "remove any digit to right" ?
Dsds7292 wrote:
Remove the far right each time. So the nine digit number should be divisible by nine. Take away the far right and the new number should be divisible by 8. Take away the next farthest digit and that number should be divisible by 7. And so on
thaithai wrote:
The number is 381654729 because
381654729 / 9 = 42406081
38165472 / 8 = 477068
3816547 / 7 = 545221
381654 / 6 = 63609
38165 / 5 = 7633
3816 / 4 = 954
381 / 3 = 127
38 / 2 = 19
3 / 1 = 3


How do you find this number?

Let's call the digits of the number G we are looking for respectively a1 till a9. So G = a1a2a3a4a5a6a7a8a9. For convenience we define G(1) = a1, G(2) = a1a2, ..., G(9) = G.

Step 1.
It is immediately clear that a5 = 5, because if there are only 5 digits left then we have G(5) which should be divisible by 5. That is only possible if a5 = 5.
Step 2.
An odd number is not divisible by an even number.
Hence a2,a4,a6 and a8 should be even, because they were not, G(2) would not be divisible by 2, G(4) not by 4, etc.
So a2,a4,a6,a8 = 2,4,6 or 8.

Step 3.
Because there are 4 even numbers in G, which are used for a2,a4,a6 and a8, the digits a1,a3,a5,a7 and a9 are necessarily odd.
Using step 1, a1,a3,a7,a9 = 1, 3, 7 or 9.

Step 4.
G(6) has to be divisible by 6. A number is divisible by six, it should be even and the sum of its digits should be divisible by 3. Since a6 is even, G(6) is even as well. G(6) is divisible by 3 if a1+a2+a3+a4+a5+a6 is divisible by 3. Because G(3) is divisible by 3 by construction, a1+a2+a3 is divisible by 3. So G(6) is divisible by 3 3 if a4+a5+a6 is divisible by 3 as well. From step 1 it follows that a5 = 5. So a4a5a6 is either 258, 456, 654 or 852. But from these four numbers, only 258 and 654 are divisible by 3.
Hence: a4a5a6 = 258 or 654.

Step 5.
Because 200, 400, 600, 800 and 1000 are divisible by 8, a number in which the second to last digit is even, is divisible by 8 if the number formed using the last two digits is divisible by 8. For example, 234216 is divisible by 8 because the twee second to last digit (2) is even, and because 16 is divisible by 8.
a6 is even. So G(8) is divisible by 8 if a7a8 is divisible by 8. a7 is an odd number, a8 even.
So the only possibilities are: a7a8 = 16, 32, 72, 96.

Step 6.
From Step 4 it follows that either a4 = 2 or 6.
If a4 = 2, then a6 = 8 (Stap 4), a8 = 6 (Step 5). Using Step 5 it follows that a7 = 1 or 9.
If a4 = 6, then a6 = 4 (Stap 4), a8 = 2 (Step 5). Using Step 5 it follows that a7 = 3 of 7.
As a consequence, a2 is either 4 or 8 .

Step 7.
A number is divisible by 3 if the sum of its digits is divisible by 3. Because G(3) has to be divisible by 3, and using that a2 = 4 or 8 (Step 6), the following possibilities arise: G(3) = 147, 183, 189, 381, 387, 741, 783, 789, 981, 987.

Step 8.
A number is divisible by 9 if the sum of its digits is divisible by 9. For the number G we are looking for this is always the case because 1+2+3+4+5+6+7+8+9 = 45, which is divisible by 9.
Using the results of the previous steps, we find the following possibilities for G. 147258963
183654729
189654327
189654723
381654729
741258963
789654321
981654327
981654723
987654321

We did not check yet wether G(7) is divisible by 7. The possibilities above can easily be checked one by one. It turns out that from these numbers only G(7) = 3816547 is divisible by 7.

So we found the number, there is only one possibility and it is G = 381654729.
thaithai wrote:
i don't change anything the answer.i want u know i just copied and pasted.
Dsds7292 wrote:
So you copyed and pasted the answer. Did you figure it out on your own?
Hoodlum wrote:
what's (click to show)
Warrant ☰ ★Officer I and a Gentleman
AlexCheckMate wrote:
Don't look if you don't want to know. :<
Spoiler (click to show)
“Gravitation cannot be held responsible for people falling in love. How on earth can you explain in terms of chemistry and physics so important a biological phenomenon as first love? Put your hand on a stove for a minute and it seems like an hour. Sit with that special girl for an hour and it seems like a minute. That's relativity.”

― Albert Einstein
elysium5 wrote:
It's not
Spoiler (click to show)
?
"I can picture in my mind a world without war, a world without hate. And I can picture us attacking that world, because they'd never expect it."