If the polynomial X - (A-1) x to the third power + 5 * x to the second power - (B + 3) X-1 does not contain the third power and X term of X, then what is the value of a + B Why B = - 2? What is the relationship between 1 - (B + 3) = 0 and 1 Why not B + 3 = 0

If the polynomial X - (A-1) x to the third power + 5 * x to the second power - (B + 3) X-1 does not contain the third power and X term of X, then what is the value of a + B Why B = - 2? What is the relationship between 1 - (B + 3) = 0 and 1 Why not B + 3 = 0

The third power of X - (A-1) x + the second power of 5 * x - (B + 3) X-1
=-(A-1) the third power of X + the second power of 5 * x + [1 - (B + 3)] X-1
If there is no cubic power and X term of X, then their coefficient is 0
So - (A-1) = 0, 1 - (B + 3) = 0
a=1,b=-2
So a + B = - 1

The steps of finding the value of ^ ax + B = A-2 + B-3 are clear

A-B=(x^2-x+b)-(x^2-ax+3)=(a-1)x+(b-3)
In order to make (A-1) x + (B-3) = x + 2, the coefficients of each term must be equal, that is, the coefficients of the primary term and the constant term must be equal respectively
a-1=1,b-3=2
a=2,b=5

It is known that the polynomial (a-b) x ^ 4 + (B-1) x ^ 3 - (A-2) x ^ 2 + ax-4 about X does not contain x ^ 3 and x ^ 2 terms. Try to write this polynomial and find out the value when x = - 2

The coefficient of x ^ 3 and x ^ 2 is 0
That is, B-1 = 0, A-2 = 0
So, a = 2, B = 1
The original polynomial is x ^ 4 + 2X-4
So, when x = - 2, the value of polynomial = 16-4-4 = 8

Xiaohua said: "among the nine natural numbers 1-9, the three digit number composed of any three consecutive natural numbers must be a multiple of 3."

Let three numbers be A-1, a and a + 1 respectively, then the sum of the three numbers is (A-1) + A + (a + 1) = 3A, which is obviously a multiple of 3

In the natural numbers of 0, 1, 2, 3 and. 9, take any three different numbers and find out the number of multiples of 3 in the three digits

Among the natural numbers 0,1,2,3 and. 9:
1. There are 0, 3, 6, 9 and 4 multiples of 3. These 4 numbers are any three different numbers, and the three digits are all multiples of 3, with a total of 3 × 3 × 2 = 18
2. Divided by 3, there are 1, 4, 7 and 3 remaining ones. The three digits of these three numbers are all multiples of 3, and there are 3 × 2 × 1 = 6
3. Divided by 3, there are 2,5,8,3 remaining 2. The three digits of these three numbers are all multiples of 3, and there are 3 × 2 × 1 = 6
To sum up, there are 18 + 6 + 6 = 30 multiples of 3

Among the natural numbers less than 10, choose any 7, at least 2 are prime. Try the drawer principle to explain the reason

The natural numbers less than 10 are 0,1,2,3,4,5,6,7,8,9
Prime number: 2,3,5,7
Total number: 4,6,8,9
Those that are neither composite nor prime are: 0,1
0,1 and other numbers do not form coprime. In the composite number, 9 and 4,6,8 form coprime, so the number 9 is not selected first, and the prime numbers form coprime, while the four numbers 2,4,6,8 do not form coprime
Drawer principle uses extreme thinking. If the condition of ensuring two numbers to be coprime is 7, then we will pick out six numbers that are not coprime, that is, 0, 1, 2, 4, 6, 8. The remaining numbers can be coprime with the number we started to pick. Therefore, at least seven numbers should be selected to make them have at least two numbers to be coprime

It is proved that for two natural numbers whose numbers are all 1, if and only if their digits are coprime, the two natural numbers are coprime

Just use mathematical induction. Just generalize the longer one
If the remainder of M divided by N is r, then GCD (M 1, n 1) = GCD (R 1, n 1)
When R and N are coprime, m and N are also coprime

Drawer principle 3: any 8 natural numbers
Drawer principle problem: any 8 natural numbers, whether there must be two numbers, their difference is a multiple of 7? Explain your reason

It's very simple. If there are seven numbers, there will be a multiple of seven
Because they are eight different numbers, the difference between the smallest and the largest is at least 7
If you don't understand, you can list eight natural numbers to see if the difference between the minimum and the maximum is at least 7

In some math cards, the numbers above are all natural numbers that are not zero, and they are all multiples of 2 or 3. Among them, cards with multiples of 2 account for 2 / 3, and cards with multiples of 3 account for 2 / 3
There are 15 cards in 3 / 4, which are multiples of 6. How many of these cards are there?
No equations,

2 / 3 + 3 / 4 - 1 = 5 / 12 cards are multiples of 2 and 3
Total cards 15 △ 5 / 12 = 36

The height of a cylinder and a cone are equal. The bottom radius is 1 decimeter. The sum of their volumes is 25.12 cubic decimeters. The volume of a cone is______ Cubic decimeter, the volume of the cylinder is______ Cubic decimeter

25.12 △ 1 + 3, = 25.12 △ 4, = 6.28 (cubic decimeter), 6.28 × 3 = 18.84 (cubic decimeter), answer: the volume of cone is 6.28 cubic decimeter, the volume of cylinder is 18.84 cubic decimeter