Question

A natural number \(y\) is defined as the sum of the digit of the number \( x \), and \( z \) as the sum of the digits of the number \( y \). How many natural numbers \( x \) satisfy the equation \( x+y+z = 60\)?

Answer 1

wth!?!?! i am so not getting the solution

Answer 2

lol :D you asked for a hard one ;)

Answer 3

lol i am not counting it this time LOL

Answer 4

x must be less than 60, right?

Answer 5

so we can count! :-D

Answer 6

There you go IShaan :D

Answer 7

intuition mwahahaha

Answer 8

where'd it go?

Answer 9

What if the question was x + y + z = 1000...

Answer 10

You don't need to be so happy :P

Answer 11

i won't even dare to solve it then

Answer 12

is it called the digital root or sth like that?

Answer 13

program a program and let the computer do it

Answer 14

yes exactly, as my teacher would say this CAT/GRE level problems are generally not meant to be generalize but only particularize.

Answer 15

*these aren't digital roots - search definition in google :)

Answer 16

anyone still counting?

Answer 17

wait
i think i have an answer

Answer 18

I was answering to Ishaan.

Answer 19

lets say there is some x y z that satifisfies x+y+z = 60.

Answer 20

Please no hints!

Answer 21

adding 0s doesn't affect the answer.

Answer 22

i think it's ok to count LOL may be there are not many numbers.

Answer 23

44, 47

Answer 24

so if we add 0s in x (or write), that doesn't really affect the answer

Answer 25

how about 440, 470, 404, 407, 704, 70010000002001?

Answer 26

have i done it wrong(ly)?

Answer 27

that's a way to get around the problem :D

Answer 28

Let \(x=10a+b\), where \(a\le 5\) , \(b\le 4\).
Case1: \(a+b\le 9\), then
\[x+y+z=10a+b+a+b+a+b=60 \implies 4a+b=20\]
The integers solutions are (5,0), (4,4), (3,8).

Answer 29

and yeah 50 too

Answer 30

Our first three solutions are \(x=50, 44\) and \(38\).

Answer 31

oh 38 as well

Answer 32

Only 1 left.

Answer 33

50,44,47,38

Answer 34

it turns out that im wrong :D:D:D:D:D:D:D:D

Answer 35

Magnum opus Ishaan & Mr.Math :)

Answer 36

what makes you think of this question???

Answer 37

hmm lollylau i think it's more about your approach towards the problem, you need a simplistic approach for gre i guess you can't make it complicated and let it eat up your time

Answer 38

what should be the last number?? hmmmmmmmmm

Answer 39

program:
enter range x to y
enter sum s
save x as x
save y as y
A
if x>y, stop
add up all the numbers of x(value of n*10^x,...n.)
save it as b
add up again
save as c
add a b c
if a+b+c=s, jot it down (in some way)
x=x+1
goto A
if x

Answer 40

the last bit 'if x' shouldn't be there.

Answer 41

i meant add up x b c

Answer 42

...

Answer 43

31 numbers ,0 、1、2、3、..........30right?

Answer 44

38 is not correct, I made a mistake up there.

Answer 45

My apologies, there is actually 3 solutions of this form.