1.3 Workshop 3 (Construction)

In this workshop, work on solving all of the following seven problems. You are encouraged to discuss the problems with those in your group but DO NOT work on writing the proofs up together.

•

Your assessed problem H01 is Problem 3. You must solve this problem, write up its proof and submit it on Gradescope as part of your portfolio assessment by the deadline stated below. Your write-up of the assessed proof must be your OWN work.
•

More guidance on submitting and resubmitting your proofs can be found on IMU Learn $\rightarrow$ Assessments.
•

You will be expected to talk and answer questions about your solutions to the remaining problems with your group in the Communication workshop next week.
•

You are free to use any of the results in the summary notes in your proofs if they are relevant to your solution.

Important Dates

•

DEADLINE FOR ASSESSED PROOF H01: 10:00AM Tuesday 23rd September 2025
•

H01 Feedback returned: 10:00AM Monday 29th September 2025
•

H01 Resubmission deadline: 10:00AM Tuesday 7th October 2025
•

H01 Final deadline: 10:00AM Friday 28th November 2025
•

H01 RESIT DEADLINE: 10:00AM Friday 24th April 2026

1.3.1 Problems

1.

Let $a,c\neq 0$ , $b$ and $d$ be integers. Prove that, if $a|b$ and $c|d$ , then $ac|bd$ .
2.

H01 Resit: Prove that, for every pair of non-zero integers $a$ and $b$ , the numbers

$\frac{a}{\gcd(a,b)}\text{ and }\frac{b}{\gcd(a,b)}$

are coprime.
3.

H01: If $p$ and $q$ are distinct primes and $n$ is an integer such that $p|n$ and $q|n$ , prove that $pq|n$ .
4.

Let $a$ , $b$ and $c$ be integers with $a$ and $c$ coprime. Prove that if $c\mid ab$ then $c\mid b$ .
5.

Prove that $\sqrt{13}$ is not rational.
6.

Prove that $\sqrt{15}$ is not rational.
7.

Prove that $\log_{2}(3)$ is not rational.