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 71
73 79 83 89 91
2：Perfect number:
Let P is the prime 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:
153 370 371 407
Let is four digital integer .if
Then the number is called Roses number. There are only three numbers:
1634 8208 9474
4：The Fundamental Theorem of Arithmetic
Every natural number n can be written as a product of primes uniquely up to order.
n=i=1kpiri
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 a, b, c, d, m ,k∈Z (k>0, m≠0).
If a≡b mod m, c≡d mod m 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, when
9：Palindrome, such as 83438, is a number that remains the same when its digits are reversed.
There are some number not only palindrome but 112=121，222=484，114=14641
(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