AMC10美国数学竞赛讲义.docx

AMC中的数论问题1：Remember the prime between 1 to 100：2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 7173 79 83 89 91 2：Perfectnumber:Let is the prime number.if is also the prime number. then is the perfect number. For example:6,28,496.3: Let is three digital integer .if Then the number is called Daffodils number. There are only four numbers:153 370 371 407Let is four digital integer .if Then the number is called Roses number. There are only three numbers:1634 8208 94744：The Fundamental Theorem of Arithmetic Every natural number can be written as a product of primes uniquely up to order. 5：Suppose that a and b are integers with b =0. Then there exists unique integers q and r such that 0 ≤ r |b| and a = bq + r.6：(1)Greatest Common Divisor: Let gcd (a, b) = max {d ∈ Z: d | a and d | b}.For any integers a and b, we have gcd(a, b) = gcd(b, a) = gcd(±a, ±b) = gcd(a, b ? a) = gcd(a, b + a). For example: gcd(150, 60) = gcd(60, 30) = gcd(30, 0) = 30 (2)Least common multiple:Let lcm(a,b)=min{d∈Z: a | d and b | d }. (3)We have that: ab= gcd(a, b) lcm(a,b)7：Congruence modulo n If ,then we call a congruence b modulo m and we rewrite .(1)Assume.If then we have , , (2) The equation ax ≡ b (mod m) has a solution if and only if gcd(a, m) divides b. 8：How to find the unit digit of some special integers(1)How many zero at the end of For example, when, Let N be the number zero at the end of then(2)Find the unit digit. For example, when9：Palindrome, such as 83438, is a number that remains the same when its digits are reversed. There are some number not only palindrome but (1)Some special palindrome that is also palindrome. For example :(2)How to create a palindrome? Almost integer plus the number of its reversed digits and repeat it again and again. Then we get a palindrome. For example:But whetheranyintegerhasthisPropertyhasyet to prove(3) The palindrome equation means thatequation from left to right and right to left it all set up.For example： Let and