problem to find out the numbers

Hi

My son has brought the below problem from his school and unable to achieve the solution. please advise if any one could help how to resolve this.

Problem: Four positive integers are arranged in a 2 X 2 table. For each row and column of the table, the product of the two numbers in a row or column is calculated. When all four such products are added together, the result is 1001. What is the largest possible two numbers in the table that are neither in the same row nor in the same column?"

thanks

Re: problem to find out the numbers

I would label the 4 integers, from left to right, top to bottom as $\displaystyle a,b,c,d$ and so we have:

$\displaystyle ab+cd+ac+bd=1001$

$\displaystyle a(b+c)+d(b+c)=1001$

$\displaystyle (a+d)(b+c)=1001=7\cdot11\cdot13$

Now the two factors on the left are the sums of two numbers that are diagonal with respect to one another, and are therefore neither in the same row nor in the same column. The largest possible sum for one of these pairs (let's use the first) is:

$\displaystyle a+d=11\cdot13=143$

If we want both numbers to be as large as possible, then one could be 72 and the other 71. If we want just 1 of the numbers to be as large as possible, then one could be 143 and the other 1.