Question

Let a secret three digit number be cba.
If the sum of cab + bac + bca + abc + abc = 2536, what is cba ?

Answer 1

abc+abc??

Answer 2

it was in the question..

Answer 3

We should let the number be secret.

Answer 4

nice idea

Answer 5

no, open the secret :D

Answer 6

The sum of all six numbers is 222c + 222b + 222a
If u omit cba = (100c + 10b + a), what is left is
N = 122c + 212b + 221a which is what we are given.

Observing 221 = 13*17,
we have N = 5c + 4b mod 13 and N = 3c + 8b mod 17
(u could have used factors of 122 or 212, but 2*61 and
2*2*53 are not nearly so convenient as 13 and 17.)

For N = 2536, we have N mod 13 = 1 and N mod 17 = 3
So u have 1 = 5c + 4b mod 13 and 3 = 3c + 8b mod 17

Since b and c are digits in range 0-9, u can try each value of b for 0 through 9 .

Compute 4 x b and 4 x b mod 13.
Subtract that from 1 to (5c mod 13).
Then find a suitable value for c which satisfies that.
For example 5c = 1 mod 13 -> 5c = 40, c = 8.
compute 3c+8b (mod 17) and find the one which equals 3..

we see it is the row where b = 7 and c = 5.
So we know that the original number was 57a.
To determine a, we take 2536 - 122*5 - 212*7 = 2536 - 610 - 1484 = 442
Dividing by 221, we get a = 2, so the original number was 572.
We can verify that 527+725+752+257+275 = 2536. ..

Answer 7

fun will be gone

Answer 8

hm.. i guess.. this is easier method:
2536 +/-multiple of 99=multiple of 111

Answer 9

vishal is a fraud , he finds answers online

Answer 10

here is the banana problem, compare http://www.gottfriedville.net/mathprob/misc-bananas.html

Answer 11

HAHA, lol

Answer 12

copy, paste?

Answer 13

or typed entirely?

Answer 14

he copied and pasted the whole thing. he is a fraud

Answer 15

ok, lets banish him xD

Answer 16

whats the challenge in googling?

Answer 17

no challenge, probably the question-poster is making a fun of us :/

Answer 18

well the question is fine, i dont like people who just post the whole answer. i want to work on it

Answer 19

well, maybe its not a big deal. ok , forget about it

Answer 20

better have the answer than not, right?

Answer 21

i agree ...

Answer 22

ok im angry that he gets a medal for it

Answer 23

a medal for using google?

Answer 24

it's all right ... medal's just nothing

Answer 25

but i gave him a medal, can i take it back? xD

Answer 26

i became his fan..

Answer 27

still ... i got to see a nice question and answer.

Answer 28

un-fan :D

Answer 29

hehe

Answer 30

lol ... seriously i don't like the word fan ... i believe we all have equal abilities .. some use and some don't use ... but i think it's fun

Answer 31

but he can do it himself too..

Answer 32

this one is do-able

Answer 33

he says, i have all the answers on a file. yeah its called GOOGLE!!!

Answer 34

:D

Answer 35

cab + bac + bca + abc + abc = 2536
100c+10a+b+100c+10a+c+100b+10c+a+2(100a+10b+c) = 2536
213c +221b + 221a =2536
maybe this is good start?

Answer 36

i still dont understand the gorilla problem

Answer 37

ok, if u would replace last abc by cba, the result would be different, the sum would b divisible by 222
and the difference between them is divisible by 99
then, i guess hit and trial method

Answer 38

the gorilla cant understand how he can solve that too xD

Answer 39

213c +(a+b)221 = 2536
but this gives more then one solution if 221 is more than once

Answer 40

and it does

Answer 41

cab + bac + bca + abc + abc = 2536
100c+10a+b+100c+10a+c+100b+10c+a+2(100a+10b+c) = 2536
213c +221b + 221a =2536
maybe i made a mistake here somehere...

Answer 42

\[cab+bac+bca+2abc=2536\]\[100∗(2a+2b+c)+10∗(2a+2b+c)+(a+b+3c)=2536\]\[a+b+3c=10x+3\tag1\]\[x+2a+2b+c=10y+6\tag2\]\[y+2a+2b+c=25\tag3\]
From (1), (2) and (3),
\[x−11y=−22\]
I think it would be safe to assume x=0 and y=2.\[2a+2b+c=23,a+b+3c=6\]We all know 0<a,b,c<9.
So, it wouldn't be that hard to guess all the numbers.

Answer 43

No :/ this isn't right