Let f z z z be the function defined by fab gcdab b where z...

Let f: Z^(+) × Z^(+) -> Z be the function defined by f((a,b)) = gcd(a,b) + b where Z^(+) is the set of positive integers and gcd(a,b) is the greatest common divisor of a and b. Enter the value f((10,4)) = Is f one-to-one? Yes No What is the image of f? Z^(+) and x ≡ 1 mod 4 Z × Z Z and x ≥ 2 None of the above Let f: Z+Z+ -> Z be the function defined by f(,b) = gcd(a,b) + b where Z+ is the set of positive integers and gcd(a,b) is the greatest common divisor of a and b. Enter the value f((10,4)) Is f one-to-one? Yes No What is the image of f? Z+ {x: x ∈ Z and x ≡ 1 mod 4} Z × Z Z {x: x ∈ Z and x^2} None of the above

Transcript

00:02 Hello, in this question we have to answer or do some exercises in modular arithmetic.

00:09 The first is we have to find the gcd of these two numbers using euclidean algorithm.

00:15 Now 320 divided by 24 is 1, 24, 80, 24, 3 is 70, 24 times 2 is 48, 3 is 72.

00:34 So it is 8.

00:38 So gcd 24, 320 is equal to gcd 8 and 24.

00:52 And gcd 8 and 24 is 8.

00:56 So this thing is 8.

01:01 Second question is if gcd of abc is 12, what is gcd of abc and 16? well gcd of abc and 16 is equal to gcd of gcd of abc and 16.

01:22 Gcd of these four numbers is equal to gcd of the first three numbers and 16.

01:31 And it is given to be gcd of, this gcd is given to be 12 and gcd of 12 and 16 is 4.

01:48 Gcd of these four numbers is 4.

01:52 Now the third question is if n is non -negative, what is gcd of 2n plus 1 and n? well gcd of 2n plus 1 and n is equal to gcd of 2n plus 1 minus n and n.

02:12 This thing is gcd, sorry let us do 2n plus 1 minus 2n.

02:19 Yeah let me erase it and write properly.

02:30 Is equal to gcd of 2n plus 1 minus 2 times this thing comma n.

02:42 Now 2n plus 1 minus 2n is 1 which is equal to gcd of 1 and n.

02:50 And gcd of 1 and n is 1.

02:53 This thing is 1.

02:55 Now gcd of 2n, 291 and 42.

03:01 Well okay so 291 let us divide by 42.

03:07 Now 42 times 7, let us see what it is, 2 times 7 is 14, 1.

03:21 4 times 7 is 28, 1, 29.

03:26 So it would be 6.

03:29 Let us erase.