Required Mathematical Intelligence

Chapter 90
7.11 is the name of a chain store in the United States, which deals in food and some daily necessities.One day, the general manager of the store asked a question. He asked: "There is a customer who bought four small commodities. The price of these four commodities added up to $7.11, and the product of the prices of these four commodities was exactly $7.11. $[-], what are the prices of these four items, please?"

[Answer: According to the original question, such an indeterminate equation can be written:
ABCD=7.11
A×B×C×D=7.11
The indeterminate equation consists of two equations and has four unknowns, and the solution of the unknowns cannot be obtained by the general method of solving equations (this is why this kind of equation is called an indeterminate equation).To solve indeterminate equations, it is necessary to use the explicit or implicit conditions given in the title to assist in the solution.

People are not used to working with decimals, so this equation can be converted to integers:

ABCD=711
A×B×C×D=711000000
首先，要从这711000000着手，711000000等于79×5×5×5×5×5×5×3×3×2×2×2×2×2×2，ABCD必定分别是它们某几个之间相互的乘积。这里隐含的已知条件是：ABCD，均是正整数，在数值在1到711间（确切地说，ABCD每个数都不小于1，不大于708）。

Note that among the multipliers decomposed above, the more prominent number is 79, which only appears once and is the largest. It is the most obvious target in solving the case.In ABCD, one of them must contain 79 (which is a multiple of 79).Because we said above that any number of ABCD, including the number containing 79, cannot be greater than 711, then there are 79 possible values ​​that the number containing 711 is less than 6, from large to small are 79×3×2=474 , 79×5=395, 79×2×2=316, 79×3=237, 79×2=158, and 79 itself.Look, we have narrowed the scope of detection to six numbers at once, and the number containing 79 in the answer to this question is among these six numbers.

Let's take a look at each of these six possible numbers and see if they can satisfy the requirements of being the solution of the equation.

第一个，看看474.711000000除掉474（79×3×2）后，剩下的数是5×5×5×5×5×5×3×2×2×2×2×2，这些数字要组合成三个数，这三个数的和要等于711-474=217.我们知道，由乘数分别组合来的几个数，在它们数字最接近时，其和最小。例如，2×2×2×2×2×2组合成两个数字时，只有在组合成2×2×2和2×2×2时，它们的和最小，为16，其它的任何组合成两数的和，都大于16（例如，2×2×2×2 2×2=20）。我们可以看到，5×5×5×5×5×5×3×2×2×2×2×2能组合成的和为最小的三个数（最为接近的三个数）是100，120，125，而它们的和是345，大于所要满足的217.因此，无论它们如何组成三个数，都只可能大于217，而不可能满足等于217的作案条件/解题条件，那么问题出在哪里呢？问题出在，79×3×2=474不可能是该题的解，即474不是ABCD中的任何一个，因为如果ABCD其中一个是474，其它数无论如何组和，都不可能满足那两个方程式。这样，我们可以排除474.
Second, look at 395 (79×5).Using the same analysis, we can see that after 711000000 is removed from 395, the remaining numbers, the three numbers that can form the smallest sum are 120, 120, 125, and the sum is 365, which is greater than the desired 711-395=316 . By the same token, 395 can also be ruled out of suspicion.

The third one, look at 316 (79×2×2), and of course use the same analysis method.Ha, guess what will happen this time?Hehe, our luck was really good this time, the flowerpot on the balcony accidentally fell and hit the head of the thief downstairs who was about to break into the house by breaking the window. After 711000000 is removed from 316, the sum of the remaining arrays is the smallest three numbers, 120, 125, 150, and 120 125 150=395 is exactly equal to 711-316. As a result, when we were excluding suspects, we accidentally hit the right one. Caught the guy who was committing the crime, 316, 120, 125, 150 are just a set of solutions that meet the conditions of the original question.Moreover, in the case where a number is 316, except for 120, 125, and 150, the other three numbers combined must be greater than 395. Therefore, in the case where a number is 316, there is only this group of solutions.

Catch a group of criminals, but are there other criminals?In mathematical language, is this set of solutions the only solution?

Of the six possible values ​​containing 79, we analyzed three, leaving three.We also need to check the remaining three numbers.

Fourth, look at 237 (79×3).This time, using the above method will not work, because the minimum sum of the three numbers below, after being divided by 711000000, can be less than the difference between 711 and the number.This time, we use a new method.If one of the four numbers is 237, then the sum of the remaining three values ​​should be 711-237=474. Let's look at 711000000 divided by 237, we get 5×5×5×5×5×5×3 ×2×2×2×2×2×2, pay attention to six of them 5. If these three values ​​contain 5, then their sum must also be divisible by 5.But 474 is not divisible by 5, indicating that at least one value does not contain 5. Is it possible that only one value contains 5?We see that the multiplication of six 5s is equal to 15625, which is far greater than the required sum of the three values ​​of 474, so these six 5s cannot be completely in one value.Likewise, it is impossible to have five 5s multiplied together (to get 3125), or four 5s to be multiplied together (to get 625) in a single value.Therefore, the only possible situation is that among the two numbers containing 5, there are three 5s in one number, and three 5s in the other number. In this way, these two numbers can only be 125 or 125×2 (impossible is 125×3 because 125×3 125 is greater than 474).Therefore, we only have two sets of possible values, one is 125, 125, 192, and the other is 125, 250, 96. The sum of these two sets of values ​​is not 474, and neither of them is our solution.exclude!
Fifth, look at 158 ​​(79×2). 158 is also not divisible by 5, so we can still use the above method.The process is not wordy, and the possible four sets of values ​​​​are 125, 125, 288; , so, 125 is also innocent.

The sixth and last one, see that 79.79 is also not divisible by 5, we can follow the example, omit the process, and get six sets of values, which are: 125, 125, 576; 125, 250, 288; 250, 250, 144; 125, 500, 144; 125, 375, 192; 250, 375, 96. We are happy to see that none of them meet the requirements (the sum of the three must be equal to 711-79), so 79 is also Innocent.

Looking back, among the six possible values ​​including 79, only 316 is satisfied, and a set of solutions has been found, 316, 120, 125, 150, and it is the only set of solutions.

Don't forget, for the convenience of calculation, we removed the decimal point, and we have to add the decimal point back.

Final answer: The prices of these four items are: $3.16, $1.20, $1.25, and $1.50. ]
(End of this chapter)