Suppose that f(x)is a polynomial with integer coefficients, such thatf(n)is a multiple of n, for all positive integers n.What is the constant term of f(x)
To find the constant term of the polynomial f(x), we need to consider the property given: f(n) is a multiple of n for all positive integers n.
Let's start by considering a value of n that is prime. Since f(n) is a multiple of n, we can write f(n) = n * q, where q is some integer. Since n is prime, q must be an integer multiple of n.
Now, let's consider a prime power n^k, where k is a positive integer. Since f(n^k) is a multiple of n^k, we can write f(n^k) = n^k * r, where r is some integer. This means that f(n^k) is divisible by both n^k and all of its factors, which are powers of n (n, n^2, n^3, ...).
From this, we can conclude that any polynomial with the given property will have a constant term that is a multiple of the product of all prime powers. In other words, the constant term of f(x) must be a multiple of the least common multiple (LCM) of all prime powers.
Therefore, the constant term of f(x) is a multiple of the LCM of all prime powers.