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It is proved that M = 2006 ^ 2 * 2007 ^ 2 + 2006 ^ 2 + 2007 ^ 2 is a complete square number
m=(2007-1)^2+[(2007-1)2007]^2+2007^2
=2007^2-2*2007+1+(2007^2-2007)^2+2007^2
=2007^4-2*2007^3+3*2007^2-2*2007+1
=2007^4+2*2007^2+1-2*2007(2007^2+1)+2007^2
=(2007^2+1)^2-2*2007(2007^2+1)+2007^2
=(2007^2+1-2007)^2
Let m = 1! + 2! + 3! + 4! + ····· + 2003! + 2004!, then what is the sum of the last two digits of M?
In fact, we don't need to consider the sum of 10! To 2004! Because their last two digits must be 00.1! + 2! + 3! + 4! + 5! + 6! + 7! + 8! + 9! = 1 + 2 + 6 + 24 + 120 + 720 + 5040 + 40320 + 362880 = 409113, so the sum of the last two digits is 4
From 1, 2, 3 Of these natural numbers, 1988 and 1989, the most can be taken out______ So that the difference between the two numbers is not equal to 4
Put one, two, three The 1999 numbers are divided into four groups with tolerance of 4, 1, 5, 9, 13 19831987 --- a total of 497 numbers; 2, 6, 10, 14 1984, 1988 - 497 numbers in total; 3, 7, 11, 15 A total of 1987, 498, 988, 16, 519, 987, 987, 987, 987, 987, 987, 987, 987, 987, 987, 987, 987, 987, 987, 987, 987, 987, 987, 98 19821986 --- a total of 496 numbers; we found that: 1. The difference between any two adjacent numbers in each row of four rows is 4, and the difference between two non adjacent numbers can not be 4; 2. The difference between any two numbers belonging to two different rows can not be 4, because if the difference is 4, the two numbers will be classified into one row, which is obviously contradictory to the fact; so we use this method to select the number that meets the requirements: the first three Choose one row every other number, and each row can have 249 numbers at most. In the fourth row, choose 4 first, and then choose one every other number, 249 numbers can be selected, and finally 249 × 4 = 996 numbers can be obtained. Answer: you can take 996 numbers at most, so that the difference between each two numbers is not equal to 4
RELATED INFORMATIONS
- 1. All natural numbers are integers, and all integers are natural numbers______ (judge right or wrong)
- 2. 0 is both an integer and a natural number______ Judge right or wrong
- 3. The concept of natural number
- 4. There is an operation program, when a ⊕ B = n (n is a constant), define (a + 1) ⊕ B = n + 1, a ⊕ (B + 1) = n-2, now known 1 ⊕ 1 = 2, then 2010 ⊕ 2010=______ ．
- 5. There is an operation program, can make: a ⊕ B = n (n is a constant), get (a + 1) ⊕ B = n + 1, a ⊕ (B + 1) = n-2, now known, 1 ⊕ 1 = 2, then 2013 ⊕ 2013=______ ;2014⊕2014=________
- 6. There is an operation program that can make (a + 1) ⊕ B = n + 1, a ⊕ B + 1 = n-2 when a ⊕ B = n (n is a constant). Now we know that 1 ⊕ 1 = 2, then 3 ⊕ 3=______ ．
- 7. There is an operation program, which can make a ⊕ B = n (n is a constant), get (a + 1) ⊕ B = n + 1, a ⊕ (B + 1) = n-2=______ ．
- 8. There is an operation program that can make (a + 1) ♁ B = n + 2, a ♁ (B + 1) = n-3, 1 ♁ 1 = 42009 ♁ 2009 =?
- 9. m. N is a non-zero natural number, m × 25 of N & lt; m, m × 22 of N & gt; m, find the value of n!
- 10. If the quotient of natural number a divided by natural number B is 9, then the greatest common factor of a and B is () reason
- 11. F (x) defined on a set of positive integers is a positive integer for any m, N, where f (M + n) = f (m) + F (n) + 4 (M + n) - 2, and f (1) = 1 [1]
- 12. Let f (x) = ax ^ 2 + BX + C, where a is a positive integer, B is a natural number, and C is an integer If the inequality 4x ≤ f (x) ≤ 2 (x ^ 2 + 1) holds for any real number x, and there exists x0 such that f (x0)
- 13. If x ^ 2 + ax-9 = (x + m) (x + n) m, n is a positive integer, then the value of natural number a is? A.0, - 8.8b. Plus or minus 8 c.0, - 8 d.0,8 d.0
- 14. The result of (X & sup3; + MX + n) × (X & sup2; - 5x + 3) does not contain the terms of X & sup3; and X & sup2; to find the value of M, n
- 15. Given that the equation (m-2) x ^| M-1 | + (n + 3) y ^ (n-8) = 6 is a quadratic equation of two variables, find the value of M and n It's a process
- 16. Given that X and y are natural numbers and satisfy XY + X + y = 11, we can find the value of X and y
- 17. Given that x, y are natural numbers, x > y and satisfy (x + y) + (x + xy-y) + X / y = 243, find the value of X + y
- 18. a. B and N are fixed natural numbers, and for any natural number k (K ≠ b), A-K ^ n can be divisible by B-K, it is proved that a = B ^ n
- 19. Let every element of a nonzero square matrix of order n be equal to its algebraic cofactor, and prove that R (a) = n
- 20. It is known that for any natural number m, N, f (m) f (n) = f (n + m) + F (n-m) It is known that for any natural number m, N, there is f (m) f (n) = f (n + m) + F (n-m), where f (0) ≠ 0, f (1) = 1, find f (n) ———————————————————————————————————————— -------------------------- Sorry, cos n π