AMC10美国数学竞赛讲义

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 number.if

21p - is also the prime number. then 1

(21)2p p --is the

perfect number. For example:6,28,496. 3: Let

,0n abc a =≠ is three digital integer .if 333n a b c =++

Then the number n is called Daffodils number . There are only four numbers:

153 370 371 407 Let

,0n abcd a =≠ is four digital integer .if 4444d n a b c +=++

Then the number n 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 =∏p i r

i k

i=1

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 ,0a b mq m -=≠,then we call a congruence b modulo m and we rewrite mod a b m ≡. (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

mod a c b d m ±≡± , mod ac bd m ≡ , mod k k a b m ≡

(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 !n

For example, when 100n =, Let N be the number zero at the end of 100!then

10010010020424525125N ⎡⎤⎡⎤⎡⎤

=++=+=⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦

(2) ,,a n Z ∈Find the unit digit n

a . For example, when 100,3n a ==

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 n that 2

n is also palindrome. For example :

2222

211111211111232111111234321

11111111112345678987654321

====

=

(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:

87781651655617267266271353135335314884

+=+=+=+=

But whether any integer has this Property has yet to prove

(3) The palindrome equation means that equation from left to right and right to left it all set up.