[PDF] Answers - Mr Barton Maths PDF 111.pdf

Prove algebraically that the sum of the squares of any two consecutive numbers always leaves a remainder of 1 when divided by 4 consecutive numbes are and n

2 juil 2010 · Prove that a and c leave the same remainder when divided by n if and Remember, we only have theorem 1 1 and its corollary to work with, so we'll We now have to account for the factor of 3 which must divide p2 − q2

[PDF] Theorem If p is a prime number which has remainder 1 when

This theorem is hard to prove We can attempt to check it by listing the primes whose remainder is 1 when divided by 4 (5, 13, 17, 29, 37, 41, 53, ) and seeing if

[PDF] Solutions to Homework Set 3 (Solutions to Homework Problems

(Solutions to Homework Problems from Chapter 2) Problems from §2 1 2 1 1 Prove that a ≡ b (mod n) if and only if a and b leave the same remainder when

[PDF] Solutions to 18781 Problem Set 3 - Fall 2008 - LSU Math

(Niven 2 3 8) Find the smallest positive integer whose remainder is 1,2,3,4, and 5 when divided by 3,5,7,9, We have to prove this for any arbitrary k ∈ Z, so fix such a k an N Consider the first k primes, call them p1 = 2,p2 = 3, ,pk So the left side of the iff is equivalent to the solution in integers x, y of an equation of

[PDF] Final stages We are in the middle of two proofs of the impossibility of

will for one of the two signs not be divisible by p2 unless p divides 2f +g, but We now want to show that every prime that leaves the remainder 1 on division by

Answers to Exercises

The quotient leaves remainders of 5, 1, 4 when divided by 9, 10, 11, and so (by the (a) IT gP-1 == 1 mod p2, then replace 9 by (p + l)g and show that then

[PDF] Two formulas for the Euler ϕ-function

1 )ϕ(p a2 2 )···ϕ(p ak k ) = p a1−1 1 (p1 − 1)pa2−1 2 (p2 − 1)···pak−1 k ( pk − 1), We shall prove Theorem 1 by proving a more conceptual result below The Chinese simultaneous divisibility with remainder problems (although roughly equiv- proof that r ≡ r2 (mod n2) is very similar and we will leave it to you