Tisch
First, concept: decompose a natural number (except 0) into the form of adding several natural numbers greater than 0. Two. Type-method
1, basic type
2. Number-making type
3. Find the maximum addend
Methods: 1+2+3+ ... Close to the result but not more than the known number, and then make up the difference.
4, two-digit type
(1) and the same: the difference is small, the product is large, and the difference is large.
(2) product invariance: big difference, big difference and small difference.
5. Split number type
The product (1) is allowed to be the same: none 1 3 more and 2 less.
(2) Not allowed to be the same: continuously split 2+3+4+ ... from 2 until the target number is just exceeded.
1) Go through it several times.
2) 1 greater than 2, 1 less than 2.
extreme
Example 1. Several identical boxes are arranged in a row. Xiaoming put 42 identical balls in these boxes and went out. Xiao Cong takes out a ball from each box, then puts these balls in the box with the least balls and rearranges the boxes. Xiao Ming came back and looked at it carefully. He didn't find his friend touching the ball and the box. Q: How many boxes are there in a box?
Analysis: Suppose that the box with the least number of balls originally contained ball A, but now it has been increased to ball B, but Xiao Ming found that no one touched the ball and the box, which means that there is a box containing ball A now. This box originally contained a+1 ball.
Similarly, there is a box containing balls of a+ 1, and this box originally contained balls of a+2.
By analogy, it can be seen that there is another box containing balls of a+3, balls of a+4 and so on, so the number of balls in those boxes is some continuous natural numbers.
Now the question becomes: how many divisions are there when you divide 42 by the sum of several consecutive integers, and how many addends are there in each division?
Because 42=6×7, 42 can be regarded as the sum of 6 of 7, and:
(7+5)+(8+4)+(9+3)
It's six sixes, so:
42=3+4+5+6+7+8+9
A * * * has seven addends; Because 42 =14× 3,42 can be written as 13+ 14+ 15, a * * * has three addends;
Because 42=2 1×2, 42 can be written as 9+10+1+12, and a * * * has four addends.
Solution: There are three solutions to this problem. A * * * has 7 boxes, 4 boxes and 3 boxes.
Golden touch: skillfully using assumptions and reasoning to connect the known with the unknown.
Example 2: 1992 is expressed as the sum of several natural numbers. If you want to multiply these numbers, these natural numbers are _ _ _ _.
(1992 Wuhan Primary School Mathematics Competition)
Analysis: If an integer is divided into several natural numbers and there are numbers greater than 4, then the number greater than 4 is divided by the sum of 2 and another natural number greater than 2, then the product of this 2 and this number greater than 2 must be greater than it. However, if the split fraction contains 1, it is inconsistent with "product".
Therefore, to multiply the addend, the addend can only be 2 and 3.
However, if the addend contains three 2s, it is better to divide it into two 3s, because 2×2×2=8 and 3×3=9.
Therefore, a split natural number contains at most two 2' s, while the rest are all 3' s.
And 1992÷3=664. Therefore, these natural numbers are 664 3.
Tisso
Exercise 1. Divide 50 into the sum of 10 prime numbers, and the bigger the prime number, the better. What is the prime number of this?
2. Divide 17 by the sum of several unequal prime numbers. What is the continuous product of these prime numbers?
3. A natural number can be divided into nine consecutive natural numbers, 10 consecutive natural numbers and the sum of 1 1 consecutive natural numbers. What is the minimum number of this natural number?
4. How many different natural numbers can the number100 be written at most?
5. There are 60 bank notes, including 1 min, 1 min, 1 yuan, 10 yuan. Can the total face value of these banknotes be exactly 100 yuan?
6. There are 30 dimes and 8 nickels. How many currencies can these coins between 1 and 1 yuan be made into?
7. Is the sum of several consecutive natural numbers exactly equal to 64?
8. A number of boxes with the same appearance are arranged in a row. Xiaoming put 54 identical balls in these boxes and went out. Liang Xiao takes out a ball from each box, then puts these balls into the box with the least number of balls, and then rearranges the boxes. When Xiao Ming came back, he looked at each box carefully, but he didn't find anyone touching the ball or the box. How many boxes does a * * * have?
9. What is the sum of all natural numbers divisible by the sum of two or more consecutive natural numbers in 2000?
10. There is a ruler, which is 13cm long and has no scale. Can you draw four tick marks on it and let this ruler directly measure all the whole centimeter lengths from 1 to 13cm?