Discrete Math 1(a) (i) Prove that p and q are relatively prime, given that p =31

Question

Discrete Math
1(a) (i) Prove that p and q are relatively prime, given that p=310 and q=43. (HINT: Use Euclidean Algorithm)

       (ii) Find the GCD of 249 and 48 by using Euclidean algorithm. By using the previously found result, find
            LCM(249,48).

(b) A newly married couple decided to buy a house worth $150,000. They take a house loan from the bank for the duration of k years. The bank offers them a loan package which consists of two interest rates: the interest rate for the first two years is 1.5% and the following years is 6%. Assume they paid $550 a month to the bank. Find the implicit formula to calculate the amount of money the couple needs to pay to the bank on the n-th month

(c) Between club A and club B, club A has twice the chance to be a winner. Between club B and club C, club B has three times chance to be a winner. Club C and club D has the same chance to be a winner. What are the chances for each club to be a winner?

Explanation / Answer

Given p=310 and q=43, Euclidean Algorithm states that GCD of two numbers is same as the GCD of first number and difference of the numbers .

For two numbers to be relatively prime their GCD should be 1 ( By definition )

By euclidean algorithm

310-43=267,

267-43=224,

224-43=181,

181-43=138,

138-43=95.

95-43=52

GCD (310,43) = GCD ( 52,43 ). 43 is a prime number having 1 and 43 as factors , excluding 1 , 43 is clearly not a factor of 52 . They share only one common factor and it is 1. Their GCD is 1, hence relatively prime.

GCD(249,48)

249-48=201

201-48=153

153-48=105

105-48=57

GCD(249,48)=GCD(57,48)

factors of 48 =1,2,3,4,6,8,12,16,24,48

factors of 57=1,3,19,57

Hence GCD ( 249,48 ) is 3

We have a property Product of LCM and GCD equals Product of the numbers

There fore LCM(249,48)*GCD(249,48)= 249*48

LCM(249,48)=(249*48)/GCD(249,48)

=249*48/3

LCM(249,48)=3984

c) A=2B, B=3C, C=D

make all of them in terms of D

A=2B=2(3C)=6C=6D,

B=3C=3D,

C=D.

If they all play together A has 6 times the chance to be a winner

B has 3 times the chance to be a winner

C and D have one time the chance to be a winner

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