Two positive whole numbers are multiplied together, and the product turns out to be 11 more than their sum. What is the largest value the bigger of the two numbers can take?

Write $ab=a+b+11$ and rearrange: $ab-a-b+1=12$, which factorises as $(a-1)(b-1)=12$. The divisor pairs of 12 are $1\times 12,2\times 6$ and $3\times 4$, giving (a, b) = (2, 13), (3, 7) and (4, 5) — and nothing else. The biggest 'bigger number' is 13, and it checks out: $2\times 13=26$ while 2 + 13 = 15, a difference of exactly 11. The trap 12 reads off $a-1$ instead of a, forgetting to add the 1 back; 7 is the bigger number of the middle pair; and 11 simply echoes the constant in the question.