Question

Find the number of three-digit numbers which when divided by 11 leave a remainder of 9 and when divided by 7 leave a remainder of 2.

Answer 1

Have you heard of the Chinese remainder theorem?

Answer 2

Do you know that:

Dividend = Divisor*Quotient + Remainder

Here, we have to find Dividend that is a 3 Digit Number...

Getting or not???

Answer 3

I'll assume you haven't. Basically, all you need to know, that there is only one integer out of 77 integers that satisfy the property you want. If you look at \(n=9\), you find that this is the first such integer, since \(9=7*1+2=0*11+9\).

Hence, the integers that satisfy the desired property are of the form \(x=a\cdot77+9\). Now we need to find how many are in between 100 and 1000.

Answer 4

How do we do this? Simple. We find the number between 0 and 1000, and subtract off those in between 0 and 100.

Can you solve for \(a\) and \(m\) here? \[77\cdot a+m=1000\]

Answer 5

Don't you think @KingGeorge you are going to very difficult level to solve this question..??

Answer 6

@PUNYASRIDHARAN pls interact with the users here , dont look for worked out solution

Answer 7

@waterineyes no.

Answer 8

See, Let the Dividend be x and the quotient be p in first case when divided by 11 and q in second case when divided by 7..

So, according to Formula I have said above:

x = 11p + 9 (11 is Divisor, p is Quotient and 9 is the remainder)..

Also, x = 7q + 2 (7 is Divisor, q is quotient and 2 is the remainder)...

So, equating the above equations:

You get,

11p + 9 = 7q + 2

or,

Evaluate q from above:

\[q = (11p + 7)/7\]

Now all we have to find the least value of p such that the given expression for q becomes a whole number...

Suggest any value of p so that (11p + 7) is wholly divisible by 7..

Can you suggest????

Answer 9

Is it 7?

Answer 10

See, if put p = 0 then it becomes a whole number but it will not constitute to a 3 Digit Number...

Answer 11

So p = 7
that means our
x = 11p + 9
is actually
x = 86, which indeed,
gives a remainder of 2 when divided by 7
and gives a remainder of 9, when divided by 11.

But 86 is not a three digit number... ;)
Can you do it from here, @PUNYASRIDHARAN ?

Answer 12

@terenzreignz If we put p = 15 then what is q???

Answer 13

172/7
?

Answer 14

No no, If we put p = 14 then what is the value of q??
Can you tell me???

Answer 15

but of course :)
it's 163/7

Answer 16

Check the calculations..
It is 161 not 163 I guess so...

Answer 17

You're right, I'm sorry :)
So why did you ask me to calculate?

Answer 18

I said that the formula is:

q = (11p + 7)/7

Use this and get the value of q when p = 14..

Solve and tell me...

Answer 19

You already said it's 161, and I wholly agree
(11(14) + 7)/7
(154 + 7)/7
161/7
23

Answer 20

I meant the numerator though.... my bad :)

Answer 21

So, we are getting q a whole number..
So, p = 14 satisfies the answer...

As, x = 11p + 9 from the very first equation of mine:

So, put p = 14 here..

x = 11*14 + 9

So, The Number must be 163...

So, the three digit number is 163...

Answer 22

Yes, I get that, but that wasn't the question :)
The question was how many three digit numbers satisfy the conditions :D

Answer 23

That is also can be solved very easily:

See, here q = (11p + 7)/7

So, if we look at a closer angle to this formula then we arrive on the conclusion that if p is a multiple of 7, then we will get q a whole number..

For example, you first put p = 7 and got 86 but two digit number..

Then I have put put p = 14 (again multiple of 7), we got 163 as number..

Now put 21, 28, 35 etc to find all the three digit numbers...

Getting or not??

Answer 24

That, but my idea was
get 86 (which is not a thee-digit number)
and keep adding 77
which is the least common multiple of 7 and 11
That means adding 77 will not affect the remainder when divided by
7 OR 11.
so...
86 (not one of them)
163 (86+77)
240 (163+77)
317
394
471
548
625
702
779
856
933

And that's it, because adding another 77 yields 1010, a four-digit number
:)
@waterineyes
I got you, but I don't want to keep plugging into a linear function :)

Answer 25

So, the maximum value of p we can plug in to that equation is p = 84 because on putting 91 we are getting 4 Digit Number..

So, we can put 7, 14, 21, 35, 42, 49, 56, 63, 70, 77 and 84..

But at 7, we are getting 86 which is a 2 Digit Number, from above neglect 7..

Now, you will get a 3 Digit Number on 14, 21, 35, 42, 49, 56, 63, 70, 77 and 84..

So, how many numbers will it form:

So, there are 10 Three Digit Numbers....

Answer 26

@terenzreignz That's the direct way to do it.

@waterineyes there's actually 11 of them

Answer 27

oh there are 11, then can you please tell me what are the 11 numbers which satisfy the condition given in the question...????

Answer 28

The fastest way to do it, you don't even have to calculate the numbers. Since \[1000=12\cdot77+76\]you know that there are 13 (12+1, since we're counting 0 as well) three digit numbers that satisfy the wanted property less than 1000. Since there are 2 less than 100 (9, 86) there are \[13-2=11\]numbers.

Answer 29

If you want to see all 11 written out, terenz did a good job there
163
240
317
394
471
548
625
702
779
856
933

Answer 30

@KingGeorge
Thanks :)
I must admit, I've never heard of the Chinese Remainder Theorem until now
I'll be sure to look at it and learn it :)
Thanks again :D

Answer 31

Just as a warning, it's a little difficult to follow if you look it up on wikipedia.

Answer 32

I know that, but there are good guys out there who actually speak simple English :D
I'll seek them out...

Answer 33

Sorry, I forgot to include 28 in that sequence..

So, total 11 numbers will be there...