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# If x and y are both integers chosen at random between 1 and

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Manager
Joined: 15 Jan 2011
Posts: 101
If x and y are both integers chosen at random between 1 and  [#permalink]

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07 Apr 2012, 03:04
2
5
00:00

Difficulty:

85% (hard)

Question Stats:

50% (02:14) correct 50% (02:37) wrong based on 91 sessions

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If x and y are both integers chosen at random between 1 and 100, inclusive, what is the approximate probability that x/y is an integer?

(A) 1%
(B) 3%
(C) 5%
(D) 7%
(E) 9%
Math Expert
Joined: 02 Sep 2009
Posts: 52385
Re: If x and y are both integers chosen at random between 1 and  [#permalink]

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07 Apr 2012, 04:10
Galiya wrote:
Could someone help me with this one?what would be the least time consuming solution for the problem below?

If x and y are both integers chosen at random between 1 and 100, inclusive, what is the approximate probability that x/y is an integer?

(A) 1%
(B) 3%
(C) 5%
(D) 7%
(E) 9%

Total # of x/y possible is 100*100=10,000 (since both x and y can tale 100 values each).

Now, if:
y=1 then all 100 possible values for x will give integer value for x/y;
y=2 then all 50 even values for x will give integer value for x/y;
y=3 then all 33 multiples of 3 for x will give integer value for x/y;
y=4 then all 25 multiples of 4 for x will give integer value for x/y;
y=5 then all 20 multiples of 5 for x will give integer value for x/y;
y=6 then all 16 multiples of 6 for x will give integer value for x/y;
y=7 then all 14 multiples of 7 for x will give integer value for x/y;
y=8 then all 12 multiples of 8 for x will give integer value for x/y;

You can continue with same logic for other values of y...
...
If y is from 51 to 100 then only one value for x (namely the value when x=y) will give integer value for x/y.

Next, if you sum all those numbers: 100+50+33+25+20+16+14+12+...+50=~500.

P=~500/10,000=5%.

Or: (100+50+33+25+20+16+14+12)+50=320, values of y from 9 to 50 (42 values), inclusive would add at least 42*2=84 more numbers, so total will be at least 320+84=404, which is closer to 5% then to 3%.

Answer: C.

P.S. There is zero chances of getting something like this on actual exam, not because this question is too hard but because of tedious math you have to do (though in the middle you can apply some approximation), so I wouldn't worry about this problem at all.
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Posts: 9461
Re: If x and y are both integers chosen at random between 1 and  [#permalink]

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12 Apr 2018, 07:43
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Re: If x and y are both integers chosen at random between 1 and &nbs [#permalink] 12 Apr 2018, 07:43
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# If x and y are both integers chosen at random between 1 and

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