PUMaC 2021 · 数论（B 组） · 第 57 题

Number Theory B

1. Andrew has a four-digit number whose last digit is 2. Given that this number is divisible by 9 , determine the number of possible values for this number that Andrew could have. Note that leading zeros are not allowed.
2. The smallest three positive proper divisors of an integer n are d < d < d and they satisfy 1 2 3 d + d + d = 57 . Find the sum of the possible values of d . 1 2 3 2 5 2 3 3 5 2
3. Compute the remainder when 2 + 3 + 5 is divided by 30.
4. A substring of a number n is a number formed by removing some digits from the beginning and end of n (possibly a different number of digits is removed from each side). Find the sum of all prime numbers p that have the property that any substring of p is also prime.
5. Compute the number of ordered pairs of non-negative integers ( x, y ) which satisfy 2 2 x + y = 32045 . P e e e 3 1 2 k
6. Let f ( n ) = k . If the prime factorization of f (2020) can be written as p p . . . p , 1 2 k gcd( k,n )=1 , 1 ≤ k ≤ n k P find p e . i i i =1
7. Suppose that f : Z × Z → R , satisfies the equation f ( x, y ) = f (3 x + y, 2 x + 2 y ) for all x, y ∈ Z . Determine the maximal number of distinct values of f ( x, y ) for 1 ≤ x, y ≤ 100 . P n gcd( i,n )
8. Let f ( n ) = . Find the sum of all positive integers n for which f ( n ) = 6 . i =1 n 1