Question

Find a number which when divided by 10 leaves remainder 9, when divided by 8 leaves remainder 7, etc. down to where when divided by 2, it leaves remainder 1.

Answer 1

@bahupepe

Answer 2

@ganeshie8

Answer 3

Managed to solve it by trial and error... I don't know if there's a better way.

The divided by 2 leaves remainder 1 makes it odd
The divided by 10 leaves 9 makes its unit digit 9

Answer 4

this is a chinese remainder theorem problem

Answer 5

Aahh.. clearly out of my league. lol

Answer 6

work backwards ?

Answer 7

pff...
It's A number, and no specifications to it.
Try 10! - 1

Answer 8

The Question I am posting are from International mathematics Olympiad Level-1 sample papers given By My Coach Professor Binay Kumar Sinha

Answer 9

LOL
I'm just kidding :)

Answer 10

Or am I?

Answer 11

You're probably looking for the smallest number that satisfies your conditions, though, @goformit100
and 10! - 1
is simply ghastly XD

Answer 12

x = 9 mod 10
x = 8 mod 9
x =7 mod 8
x = 6 mod 7
x = 5 mod 6
x = 4 mod 5
x = 3 mod 4
x = 2 mod 3
x = 1 mod 2

Answer 13

I personally solved this question in this was am i correct ?

Answer 14

Okay.... time to work ...
1 divides 1
1x2 divides 1 and 2
1x2x3 divides 1,2,3,6
1x4x3 divides 1,2,3,4,6
1x4x6 divides 1,2,3,4,6,8
1x3x4x6 divides 1,2,3,4,6,8,9
1x3x4x5x6 divides 1,2,3,4,5,6,8,9,10
1x3x4x5x6x7 divides 1,2,3,4,5,6,7,8,9,10

So 2520 (that product)
would be divisible by all from 2 to 10
so take away 1
it will yield the desired remainders
2519
is your answer ;)

Answer 15

SEE

Let the number = N
So, N = 10base(a9) + 9
= 9 base(a8) + 8
= 8 base(a7) + 7
= 7 base(a6) + 6
… …
= 2abase(1) +1


i.e. N+1 is having 2,3,4,5, ….. 10 as a factor
Taking LEM of 2,3,4,5,….. 10
= 2520


is it correct ?

Answer 16

LOL @goformit100 !!!!
:)

Answer 17

Am I correct ?

Answer 18

Yup :)

Answer 19

Next http://openstudy.com/study#/updates/517bbcaae4b0249598f7a76d

Answer 20

what does LEM stand for?

Answer 21

@terenzreignz
Color me impressed!!! :-)

Answer 22

@terenzreignz wonderful logic !!!

Answer 23

Thanks, guys :)
I could summed it all up in what @goformit100 says, though (LEM, which I suspect means something along the lines of "least common multiple")

I have a tendency to overdo things :/

But a big thanks to you guys :D

Answer 24

im lost, what is the next step

Answer 25

You find the least common multiple of the numbers from 1 to 10, and take away 1 from that least common multiple.

Answer 26

x = 10 a_9 + 9
= 9 a_8 + 8
= 8 a_7 + 7
.
.
.
= 2 a_1 + 1

Answer 27

why do you take away 1 from the LCM

Answer 28

x-1 = 2*a_1
x-2 = 3*a_2
x-3 = 4*a_3

Answer 29

Because, since that number 2520 = 0 (mod <all of them>)
if you take away 1,
2519 = -1(mod 10) = 9(mod 10)
2519 = -1(mod 9) = 8(mod 9)
2519 = -1(mod 8) = 7(mod 8)
etc...

Answer 30

ok , LCM {2,3,4,5,6,7,8,9,10} = 2520

Answer 31

would it be the least number with that property

Answer 32

It would be the least.... I didn't even know what I was doing (finding the LCM) until goformit pointed it out :D

Answer 33

im not so sure i feel some numbers may exist with these properties before we reach LCM

Answer 34

2520 mod 10 = 0 mod 10 , then you subtract both sides by 1 ?
2519 mod 10 = -1 mod 10

Answer 35

and -1 mod 10 = 9 mod 10

Answer 36

2520 mod 9 = 0 mod 9
2519 mod 9 = -1 mod 9 = 8 mod 9 = 8

Answer 37

In general is it true that
x mod a = y mod a iff (x + - c) mod a = (y + - c ) mod a ?

Answer 38

Well, @ganeshie8 suppose N is the least positive integer satisfying those remainder properties...
then N+1 is a positive integer that is divisible by all from 2 to 10
Then N+1 is a common multiple of 2 to 10

Now, for N to be the "least positive integer satisfying said properties"
N+1 must be the least common multiple of those from 2 to 10

Answer 39

Seems legit, @perl

Answer 40

ok i think i see what you did now
Given
x = 9 mod 10
x = 8 mod 9
x =7 mod 8
x = 6 mod 7
x = 5 mod 6
x = 4 mod 5
x = 3 mod 4
x = 2 mod 3
x = 1 mod 2
Then
x +1 = 10 mod 10 = 0
x +1= 9 mod 9 = 0
x +1=8 mod 8 = 0
x +1= 7 mod 7 = 0
x +1 = 6 mod 6 = 0
x +1= 5 mod 5 = 0
x +1= 4 mod 4 = 0
x +1= 3 mod 3 = 0
x +1 = 2 mod 2 = 0

Answer 41

so x+1 = LCM { 2,3,4,5,6,7,8,9,10} ,
then x = LCM { 2,3,4,5,6,7,8,9,10} -1

Answer 42

hmm i got u. but the question doesnt require it to be divisible by 9 or 7 or 5 or 3

Answer 43

The question should probably say, find the smallest number?

Answer 44

It doesn't require it to be the least integer that satisfies, neither, @ganeshie8
LOL
But I <facedesked>
:)
Let's use that logic again, but only for those mentioned...
LCM{2,4,6,8,10}=120
119 then.

Answer 45

Better tell this to @goformit100

Answer 46

ya say

Answer 47

Well, as @ganeshie8 pointed out, @goformit100 your question doesn't involve any division by the odd integers from 2-10 {3,5,7,9}
So...
119 seems to be our best bet, following similar lines of thinking.

Answer 48

isnt this the question ? find the smallest positive integer x such that
x = 9 mod 10
x = 8 mod 9
x =7 mod 8
x = 6 mod 7
x = 5 mod 6
x = 4 mod 5
x = 3 mod 4
x = 2 mod 3
x = 1 mod 2

Answer 49

There's no division by 9, 7, etc. @perl

Answer 50

ohh

Answer 51

ok well we can create a second follow up question :)

Answer 52

But similar methods work. Just altering the arguments...

Answer 53

x = 9 mod 10

x =7 mod 8

x = 5 mod 6

x = 3 mod 4

x = 1 mod 2

Answer 54

ok guys Have we met with the conclusion, whoes answer is correct ?

Answer 55

119. Thank @ganeshie8 kindly :)

Answer 56

x = LCM { 2,4,6,8,10} -1 = 119

Answer 57

i dont deserve medals for this problem
-hangs head in shame-

Answer 58

finding next integer seems tricky

Answer 59

whats the next question?

Answer 60

i mean, second least integer with given props

Answer 61

Next integer would be 2<LCM> - 1

Answer 62

right

Answer 63

ohh ya :)

Answer 64

also we might want to prove that LCM is the least integer such that the respective mods produce zero.

Answer 65

anybody has any questions, i see a lot of people observing

Answer 66

You mean the least integer such that if you take away 1, you get the respective moduli?

Answer 67

Is my next question
http://openstudy.com/study#/updates/517bbcaae4b0249598f7a76d

Answer 68

http://openstudy.com/study#/updates/517bbcaae4b0249598f7a76d

Answer 69

the question just asks 'find a number' , you could even use 2*3*6*8*10 -1

Answer 70

it should say 'least positive integer'