1Please prove or disprove An integer even if and only if n2

(1)Please prove or disprove: An integer even if and only if n² is even.

(2)(a) Please use the Euclidean Algorithm to find the GCD for a=69 and b=299.

(b)Find integers m and n so that gcd(69,299)=69m+299n.

Solution

1) if n is even

n² =n*n=even * even

we know that even * even = even,

so if n is even,then n² is also even

proved

2)

a)First step: Write the larger number as the smaller number times a quotient plus the remainder.

299 = 69 x 4 + 23

Now write the smaller number as the remainder from the first step times a quotient plus a new remainder. Keep repeating this process until you obtain a remainder of 0:

69 = 23*3 + 0

This tells us that gcd(69,299) = 3