Article 1: natural number
The sum of three natural numbers is 6 larger than the smallest one, and the other is their average. The product of three numbers is 42560. Find these three natural numbers.
Analysis: First, roughly estimate, 30×30×30 = 27000, far less than 42560.40×40×40 = 64000, far greater than 42560. So the required three natural numbers are between 30 and 40.
Solution: 42560=26×5×7× 19.
=25×(5×7)×( 19×2)
=32×35×38 (to the point)
The required three natural numbers are 32, 35 and 38 respectively.
The second part: the trip to the two cities.
Two trains, A and B, leave from East and West at the same time. Car A travels 49 kilometers per hour, and car B travels 47 kilometers per hour. When they met, car A traveled 36 kilometers more than car B. Find the distance between two cities.
Answer and analysis: 36 ÷ (49-47) × (49+47) =1728 (km).
Chapter III: Drawer Principles
What happens if you put three apples in two drawers? Either put two apples in one drawer and the remaining one in another drawer; Or put three apples in the same drawer. These two situations can be expressed in one sentence: there must be two or more apples in a drawer. This is pigeonhole principle in mathematics.
The basic pigeonhole principle of quantitative relation is that if n+ 1 objects (also called elements) are put in n drawers, at least one drawer contains two or more objects (elements).
Pigeonhole principle can be summarized as follows: if there are m drawers, there are k×m+r(0
Generally speaking, if the number of elements is k times that of drawers, at least one drawer should hold (k+ 1) or more elements.
Ideas and methods of solving problems (1) Change drawers and point out elements;
(2) Put (or take out) the components in the drawer;
(3) Explain the reasons and draw a conclusion.
Example: 1 yucai primary school has 367 students born in 1999, so at least some of them have the same birthday.
One day?
Because 1999 is a running year with 366 days, it can be regarded as 366 drawers, and the 367 students born in 1999 are 367 elements. 367 elements are put into 366 drawers, and at least one drawer contains two or more elements.
This shows that at least two students have the same birthday.
It is said that human hair does not exceed 200,000. If there are 36.45 million people in Shaanxi Province, according to these data, do you know how many people in Shaanxi Province have at least the same amount of hair?
A person's hair with no more than 200,000 hairs can be regarded as 200,000 drawers, and 36.45 million people can be regarded as 36.45 million elements. Putting 36.45 million elements into 200,000 drawers, we can get.
3645 ÷ 20 = 182 ...5 According to pigeonhole principle's generalization law, we can know that k+ 1= 183.
A: At least 183 people in Shaanxi Province have the same number of hairs.
There are some balls in a bag, but the balls are different in color. Among them, red ball 10, white ball 9, yellow ball 8 and blue ball 2. Someone took out some balls with their eyes closed. How many balls does he have to take to ensure that at least four balls are the same color?
The total number of balls of four colors (3+3+3+2)= 1 1 as1"drawer", so at least (1 1) balls must be taken to ensure at least.
Answer; He must take at least 12 balls to ensure that at least four balls have the same color.