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Algebra

Blazing goIITian

 Joined: 25 Jul 2011 Post: 322
28 Mar 2012 21:36:25 IST
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6
349
Thr is an awesome answer for this one !!
Engineering Entrance , JEE Main , JEE Main & Advanced , Mathematics , Algebra

How many numbers are like 119, have 3 digits and they have this property ?

i mean if we divide 119 by 2 the remainder will be 1

and if we divide 119 by 3 the remainder will be 2

and if we divide 119 by 4 the remainder will be 3

and if we divide 119 by 5 the remainder will be 4

and if we divide by 6 the remainder will be 5

Hot goIITian

Joined: 10 Oct 2011
Posts: 120
28 Mar 2012 21:48:37 IST
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man what a question macha  :)

is it 5 ?

Blazing goIITian

Joined: 2 Mar 2012
Posts: 308
28 Mar 2012 21:50:16 IST
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Its easy, general form 119 + 60n . u get 14 such more 3 digit numbers.

Hot goIITian

Joined: 10 Oct 2011
Posts: 120
28 Mar 2012 21:51:38 IST
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i am getting 13 now in 2nd try :P

New kid on the Block

Joined: 8 Aug 2011
Posts: 14
28 Mar 2012 21:56:36 IST
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arre......all nos having this property will be 60k-1 where 60k is a multiple of 2,3,4,5 and 6.....thus there are 15 numbers....

Blazing goIITian

Joined: 25 Jul 2011
Posts: 322
28 Mar 2012 22:09:52 IST
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Kartik and Ketan is right...

Thr r 14 numbers excluding 119

@ srini : Machan :P,U got close,the second answer,

But the actual sol for this was done through mods method :

$x \equiv 1 \pmod{2}\equiv -1 \pmod 2$$x \equiv 2 \pmod{3} \equiv -1 \pmod 3$$x \equiv 3\pmod{4} \equiv -1 \pmod 4$$x \equiv 4\pmod{5} \equiv -1 \pmod 5$$x \equiv 5\pmod{6} \equiv -1 \pmod 6$
By the Chinese Remainder Theorem, we consequently have

$x \equiv -1 \pmod{60}$
Thus, the three digit numbers that have remainder $-1$ when divided by $60$ other than $119$ are $179, 269, \cdots, 959$, which are $\boxed{14}$ numbers.

Blazing goIITian

Joined: 25 Jul 2011
Posts: 322
28 Mar 2012 22:11:43 IST
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