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x is a positive integer less than 500. When x is divided by [#permalink]
05 Nov 2006, 12:47

x is a positive integer less than 500. When x is divided by 7, the
remainder is 1; when x is divided by 3, the remainder is 2. How many
such numbers are possible? _________________

Notation: A number n when divided by k has remainder r, we say that
n = r Mod k

E.g., 28 = 4 mod 6
4 = 1 mod 3
0 = n mod n

Thm: If n = r Mod k then n*t = r*t Mod k (t an integer obviously)

E.g., (for this problem)

7 = 1 mod 3 so 7*t = t Mod 3 so 7 = 1 Mod 3, 14 = 7*2 =2 Mod 3, 21 = 7*3 = 3 Mod 3 = 0 Mod 3, 28 = 7*4 = 4 Mod 3 = 1 mod 3, 35 = 2 Mod 3, ...

So the sequence is {1, 2, 0, 1, 2, 0, ...}

The question asks " How many + numbers n < 500 of the form 7*k + 1 are equal to 2 mod 3?" (so maybe it is by now clear that it's 1/3 of the numbers of this form...)

8 = (7*1 + 1) Mod 3 = 7*1 Mod 3 + 1 Mod 3 = 2 mod 3
15 = (7*2 + 1) Mod 3 = 2 Mod 3 + 1 Mod 3 = 0 mod 3
22 = (7*3 + 1) Mod 3 = 3 mod 3 + 1 Mod 3 = 1 mod 3
29 = 2 Mod 3
36 = 0 Mod 3
43 = 1 Mod 3
...
498 = (7*71 + 1) Mod 3 = 71 Mod 3 + 1 Mod 3 = 0 Mod 3

So there are 71 numbers less than 500 of the form 2 Mod 7 and 1 out of 3 is of the form 2 Mod 3 so it's just a matter of checking the endpoints and so the asnwer is 24 because we start with 2 Mod 3 and end with 0 mod 3.

That seems complicated and long-winded, but the principles can help you do these problems in no time (maybe there's a good review of modular arithmetic somewhere on the web).

Re: Number Properties [#permalink]
07 Nov 2006, 21:39

Paayal wrote:

x is a positive integer less than 500. When x is divided by 7, the remainder is 1; when x is divided by 3, the remainder is 2. How many such numbers are possible?

Vikram, 24 is answer for this question. What I was asking was what is 'x' in the question.

I agree..it is a series of numbers. But can you please post a few initial numbers in the series? _________________

The path is long, but self-surrender makes it short;
the way is difficult, but perfect trust makes it easy.

Re: Number Properties [#permalink]
09 Nov 2006, 18:54

ak_idc wrote:

Paayal wrote:

x is a positive integer less than 500. When x is divided by 7, the remainder is 1; when x is divided by 3, the remainder is 2. How many such numbers are possible?

Vikram, 24 is answer for this question. What I was asking was what is 'x' in the question.

I agree..it is a series of numbers. But can you please post a few initial numbers in the series?

x is the # of "numbers" between 2 and <500 satisfying the given conditions....

gmatclubot

Re: Number Properties
[#permalink]
09 Nov 2006, 18:54

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