Pick a number

4 6 10 14 22 26 6 9 15 21 33 39 10 15 25 35 55 65 16 24 40 56 88 104 18 27 45 63 99 117 20 30 50 70 110 130 \begin{array} { l l l l l l } & 4 & 6 & 10 & 14 & 22 & 26 \\ & 6 & 9 & 15 & 21 & 33 & 39 \\ & 10 & 15 & 25 & 35 & 55 & 65 \\ & 16 & 24 & 40 & 56 & 88 & 104 \\ & 18 & 27 & 45 & 63 & 99 & 117 \\ & 20 & 30 & 50 & 70 & 110 & 130 \\ \end{array}

Choose 6 numbers from the given table, in such a way that from each row and column, exactly 1 number is chosen.

What is the probability that the product of these numbers is divisible by 6 5 6^5 ?

If the answer is p p , give your answer as 1000 p \lfloor 1000p \rfloor .


The answer is 0.

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Vrishank Infrared
May 10, 2015

If you see the matrix carefully, each row can be viewed as a product of number 'n' and numbers 2 ,3,5,7,11,13 respectively . The number ' n' for row First is 2 For second row it is 3 For the third row it is 5 Fourth one ,it's 8 Fifth and sixth one is 9 and 10 respectively. Now if you need to select any number abiding by the conditions given The product would be 2×3×5×8×9×10×2×3×5×7×11×13. And now if one can see the highest power of 3 in the above product is 4. Our devisor is having highest power of 5. So the product would never be divisible. Hence probability would be zero irrespective of the order of selecting the numbers.

An easier way is to get the product of a diagonal in the matrix because that is the only way for a number in the matrix to be picked in a different row and column each time. Luckily, both diagonals product are the same. Thus, the product of both of the diagonals is 648648000 and when you divide it by 6^5 it is 83416.6667, which isn't an integer, and thus, the probability that the product of these numbers is divisible by 6^5 is 0%

Manu Mehta - 5 years, 11 months ago

Log in to reply

Check your thinking again as there are many other ways than the diagonals themselves.

Vinayak Verma - 5 years, 7 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...