Find the Numbers
1. Which two numbers, when added up give 1 more than when multiplied together?
2. Which two numbers give the same result when multiplied together as when the two are added?
3. Which three numbers give the same result when multiplied together as when they are added up?
4. There is a two-digit number such that if it is divided by the sum of its digits the answer is also the sum of the digits. Find the number.
1. There are many such numbers, e.g.: 3 x 1 = 3; 3 + l = 4; or 10 x 1 = 10; 10 + 1 = 11. In general, any pair of integers of which one is unity will work. This is because adding one increases a number but multiplying by one does not change it.
2. The numbers are 2 and 2. There are no other integers.
3. Multiplying 1,2, and 3 gives the same as adding them up: 1 + 2 + 3 = 6, l x 2 x 3 = 6.
4. The number we seek should clearly be a square. As among the two-digit numbers there are only six squares, then by trial-and-error method we readily find the unique solution, namely 81: 81 / (8 +1) = 8 + 1