# Extended Euclidean Algorithim - Arithmetic Modulo 60

I get the Extended Euclidean Algorithm but I don't understand what this question is asking me:

In this question we consider arithmetic modulo 60 ("clock arithmetic" on the clock with numbers {0,1,2,3,...,58,59}). Which of the following statements apply?

a. 50 X 50 = 40 modulo 60

b. 17 + 50 = 3 modulo 60

c. 4100 x 10 = 0 modulo 60

d. 1/17 = 53 modulo 60

e. 37 + 50 = modulo 60

f. 1/10 is undefined modulo 60

g. 17 - 35 = 42 modulo 60

h. 17 - 25 = 18 modulo 60

I have to pick the right ones but I'm not sure how to work it out? Like if they gave something like

17x = 1 mod(43) I could solve it but I'm not sure how you would solve the other question

P.S I have the answers I just dont want to look at them as I'd rather try to understand first as this is revision for my exam. thanks

Except for (d) and (f), these are all basically of the same form: they are all asking if the expressions on the left and right sides are equivalent mod $60$; that is, if they both leave the same remainder when divided by $60$.

(d) and (f) both ask about the multiplicative inverse of $x$ mod $60$; that is, they ask if $x^{-1}\equiv y\mod{60}$, or in other words if $xy\equiv 1\mod{60}$ (or, if $xy$ leaves a remainder of $1$ when divided by $60$).

It is a simple multiple choice question.

Hints:

$a), b), e)$ : You just have to compute and reduce to check if true ot not.

$c)$ means: ‘Is 4100\times 10 divisible by 60?’. You may have good reasons to answer without having to perform a division.

$d)$ $\iff$ 17 × 53 =1 modulo 60. Check if true.

$f)$ means ‘Is 10 a unit or not modulo 60’

$g), h)\iff$ 17 = 35 + 42, resp. 17 = 25 +18, modulo 60.