Q:

PLEASE HELP I'M VERY CONFUSED!find all natural number solutions for each congruence.1. y + 3 ≡ 2 mod 4 2. 3 - y ≡ 5 mod 9

Accepted Solution

A:
Answer: 1. y = 4n -1 2. y = 9n -2Step-by-step explanation:1. For some integer n, numbers that are congruent to 2 mod 4 are ... 4n+2. Then you have ... y + 3 = 4n +2 . . . . next, subtract 3 from both sides y = 4n -1 . . . . values of y are natural numbers for all natural numbers n___2. Similarly, ... 3 - y = 9n +5 3 - 9n - 5 = y . . . . . add y-9n-5 -9n -2 = y . . . . . . . . collect termsFor every integer n, there is also an integer -n. For our purpose, we're only interested in those values of n that make -9n be positive. Rather than require n be negative so -9n is positive, we can require n be positive and use the expression 9n for the multiple of 9.We want y a natural number, so we can write this as ... y = 9n -2 . . . . for natural numbers n