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?

Question

Komal · Accepted Answer

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 yIf 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%.