Math Project: Number Theory, Sequences, and Proofs - University

This math project provides solutions to problems in number theory. It covers finding the Greatest Common Divisor (GCD) and Least Common Multiple (LCM) using unique prime factorization. The Euclidean algorithm is used to find the GCD of two numbers. The project also explores true/false statements with proofs or counterexamples, investigates remainders of perfect squares when divided by 7, and computes terms of a sequence. Finally, it includes a proof using mathematical induction.

MATH PROJECT

TABLE OF CONTENTS
Q1.....................................................................................................................................................1
(a) Using unique prime factorisation, find GCD (270, 924) and LCM (270, 924).....................1
(b) Using the Euclidean algorithm, find GCD (132153, 4263)..................................................1
Q2.....................................................................................................................................................2
(a) Determine whether each of the following statements is true or false. If true, give a proof...2
If false, give a counter example..................................................................................................2
Q3.....................................................................................................................................................3
(a) What possible remainders do perfect squares leave when divided by 7? Recall that perfect
squares are integers of the form k 2 where k ∈ Z.......................................................................3
Q4.....................................................................................................................................................3
(a) Compute the first four terms of the sequence {(−2) k} k ≥0.................................................3
(b) Using mathematical induction, prove that for all integers n ≥ 0, 1 − 3 Xn k=0 (−2) k = (−2)
n+1..............................................................................................................................................4

Q1
(a) Using unique prime factorisation, find GCD (270, 924) and LCM (270, 924).
Ans: GCD of 270 and 924
Step 1: prime factorisation of 270
270= 2 × 3 × 3 × 3 × 5
Step 2: prime factorisation of 924
924 = 2 × 2 × 3 × 7 × 11
Step 3: Find GCD, common factor in both numbers
therefore, GCD= 2 × 3
GCD= 6.
LCM of 270 and 924
Step 1: prime factorisation of 270
270= 2 × 3 × 3 × 3 × 5
Step 2: Step 2: prime factorisation of 924
924 = 2 × 2 × 3 × 7 × 11
Step 3: Multiply each factor the greater number
LCM= 2 ×2 × 3 × 3 × 3× 7 × 11× 5= 41580
(b) Using the Euclidean algorithm, find GCD (132153, 4263).
Ans: Divide 132153 by 4263 to get result 31 and remainder 0,
where,
4263= 0*4263+31
= 31
1

Q2
(a) Determine whether each of the following statements is true or false. If true, give a proof.
If false, give a counter example.
(a) For all x, y R, [x+y] = [x]+[y]∈
Ans: if x, y R,∈
x and y are real number so that
x=1, y=2
[1+2] = [1] + [2] =3 (true)
(b) For all x R and all y Z, [x+y] = [x]+[y]∈ ∈
Ans: If x is real number and y is integer number
so that x=2, y= -1
[2+(-1)] = [2] + [-1] = 1 (True)
(c) For all irrational numbers x, the number 10x + 3 is also irrational.
Ans: assume x= √3
putting value in equation 10x+3
10√3+3 (true)
It is also irrational number.
(d) For all a, b, d Z +, if d | ab, then either d | a or d | b.∈
Ans: If Z is positive integer number
where, a=2, b= 3 and d= 1
d/ab= 1/6 then ½ or 1/3= 5/6 (False)
It is false because it generates 5/6
2

(e) For all a, b, d Z +, if d | (a + b) and d | a, then d | b.∈
Ans: Z: positive integer number
a= 2, b= 3, d= 1
d/a+b and d/a
1/5 and ½ then 1/3
It is false
Q3
(a) What possible remainders do perfect squares leave when divided by 7? Recall that perfect
squares are integers of the form k 2 where k ∈ Z.
Ans: assume that m = 7k + r and n = 7h + s,
then 7(7kh+hr+ks) + rs.
mn mod 7 = rs mod 7.
Only check the remainders for numbers from 0 to 6
7 possible remainders divide n by 7- 0,1, 2....6,
=> leave remainder-0,1,4,2,4,1
Q4
(a) Compute the first four terms of the sequence {(−2) k} k ≥0.
Ans: (-2) kk>=0:
(-2) kk>=0
Simplify:
(-2) kk: -2K2
--2K2>=0
Multiply both sides by -1
(-2K2) (-1) < = 0. (-1)
3

Simplify
2K2<=0
Divide both sides by 2
2K2/2<=0/2
Simplify
K2<=0
K<=0 and K>=0
(b) Using mathematical induction, prove that for all integers n ≥ 0, 1 − 3 Xn k=0 (−2) k =
(−2) n+1
Ans: 1-3∑ (n to k=0) (-2) ^ k= (-2) ^ n+1
Take LHS Part: -
1-3∑ (n to k=0) (-2) ^ k
let k=0 and n= 0
Putting value of k and n in LHS part
= 1
Take RHS part: -
(-2) ^ n+1
Putting n=0 in RHS part
(-2) ^ 0+1
= 1
4