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N) = p prime pn p n Functions like this, in which we can break up the value of the function at n, via the factorization of n, into the value of the function at the maximum prime powers that divide n, are called multiplicative functions. As an example, we see that the divisors of 175 = 52 71 are given by 50 70 = 1, 51 70 = 5, 52 70 = 25, 50 71 = 7, 51 71 = 35, 52 72 = 175; in other words, they can all be factored as 50 , 51 or 52 times 70 or 71 . Therefore the number of divisors is 3 × 2 = (2 + 1)(1 + 1).

A(m − 1) + b is a complete set of residues (mod m). 2. Deduce that, whenever (a, m) = 1, for all given integers b and c, there is a unique value of x (mod m) for which ax + b ≡ c (mod m). 3. Prove that if {r1 , . . rk } is a reduced set of residues mod m, and (a, m) = 1 then {ar1 , . . 4. Show that there exists r (mod b) for which ar ≡ c (mod b) if and only if gcd(a, b) divides c. 6. Linear equations in two unknowns. Given integers a, b, c can we find all solutions in integers m, n to am + bn = c ?

Where Fm = 22 + 1. 6. Show that the nth row of Pascal’s Triangle mod 2, considered as a binary number, is given by kj=0 Fnj , where n = 2n0 + 2n1 + . . + 2nk , with 0 ≤ n0 < n1 < . . < nk , when written in binary. ) References 2. Andrew Granville, Zaphod Beeblebrox’s brain and the fifty-ninth row of Pascal’s triangle, American Mathematical Monthly 99 (1992), 318-331. 2. Andrew Granville, Arithmetic Properties of Binomial Coefficients I: Binomial coefficients modulo prime powers, Canadian Mathematical Society Conference Proceedings 20 (1997), 253-275.

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