Required Mathematical Intelligence

Chapter 61

Both Sun Bin and Pang Juan are apprentices of Guiguzi; one day Guiguzi came up with this problem: he selected two different integers from 2 to 99, told Sun the product, and told Pang the sum;
Pang said: Although I am not sure what these two numbers are, I am sure you do not know what these two numbers are either.

Sun said: I really didn't know at first, but after hearing what you said, I can now confirm these two figures.

Pang said: Now that you said that, I now also know what these two numbers are.

What are these two numbers?Why?
[Answer: Suppose the numbers are X, Y; the sum is XY=A, and the product is X·Y=B.

According to what Pang said for the first time: "I'm sure you don't know what these two numbers are."From this we know that XY is not the sum of two prime numbers.Then the possible values ​​of A are 11, 17, 23, 27, 29, 35, 37, 41, 47, 51, 53, 57, 59, 65, 67, 71, 77, 79, 83, 87, 89, 95, 97...

Let's calculate the possible values ​​of B again:
The sum is the product that can be obtained by 11: 18, 24, 28, 30
The sum is the product of 17: 30, 42, 52, 60, 66, 70, 72
The sum is the product of 23: 42, 60...

The sum is the product of 27: 50, 72...

The sum is the product that can be obtained by 29...

The sum is the product of 35: 66...

The sum is the product of 37: 70...

我们可以得出可能的B为，当然了，有些数（30=5·6=2·15）出现不止一次。

At this time, after comparing and calculating according to his own numbers, Sun said, "Now I can determine these two numbers."

Based on this sentence, and the set of B we calculated, we can delete some duplicates from the set of B calculated.

The sum is the product that can be obtained by 11: 18, 24, 28
The sum is the product that can be obtained by 17: 52
The sum is the product of 23: 42, 76...

The sum is the product of 27: 50, 92...

The sum is the product of 29: 54, 78...

The sum is the product of 35: 96, 124...

The sum is the product that 37 can get:
Because Pang said: "Since you said that, I now know what these two numbers are." Then the product obtained from the sum must also be unique, and it is known from the above that only one line is left with one number, that is Sum of 17 and 52.
Then X and Y are 4 and 13 respectively.]
(End of this chapter)