When we divide a number by m, we get a remainder between 0 and m−1. The remainder depends on the number, but for squares and cubes only certain remainders are possible. These are called the remainders (or residues) of squares and cubes modulo m.

For example, if we divide any square by 3, the remainder can only be 0 or 1. If we divide any square by 4, the remainder can only be 0 or 1 as well. This idea is useful because sometimes we can prove something is impossible just by checking all possible remainders.

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Problems

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Make tables of the remainders (residues) of squares and cubes of integers modulo m for m = 3, 4, 5, 7, 8. These tables will be needed for some problems below.
Can a number of the form 200…009 (with some number of zeros between the twos and the nines) be a perfect square of an integer for any number of zeros?
Which digits can a perfect square end with?
Can the square of an integer have the form
a) 5q + 2,
b) 3q − 1,
c) 6q − 1?
Take any positive integer n. Find the sum of its digits; then find the sum of digits of that result, and so on, until you get a single-digit number R.
a) Prove that R is the remainder of n upon division by 9.
b) What values can R take if n is a perfect square?
c) What values can R take if n is a perfect cube?
Does there exist a natural number N such that:
a) N² + 1 is divisible by 3?
b) N³ + 3 is divisible by 99?
Prove that if x² + y² is divisible by 3 (with x, y integers), then x and y are both divisible by 3.
Can the sum of the squares of two odd numbers be a perfect square? What about the sum of the squares of three odd numbers?
Let a, b, c be natural numbers with a + b + c divisible by 6. Prove that a³ + b³ + c³ is also divisible by 6.

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Where do you hold your classes?

We hold our classes online or on-site on Saturdays at our branch in Pimlico Academy, London.
You can find our timetable here.

What do you need to start learning online?

For lessons you only need a computer or phone with a microphone, camera and Internet access. Wherever you are - in London, Nottingham, New York or Bali - online lessons will be at hand.

When can I take the trial lesson?

You can get acquainted with the school at any time convenient for you. To do this, just leave a request and sign up for a lesson.

What should I expect from the trial lesson?

The trial lesson is a 30-minute online session designed to get a sense of how your child approaches mathematical thinking and problem solving. (In practice, it often runs a bit longer if the student is engaged!)

We typically explore a range of fun and challenging problems drawn from competitions. We adapt the difficulty based on how the student responds, aiming to make it both accessible and stimulating.

After the session, we’ll have a quick conversation with the parent to share observations and suggest a personalised path forward.

I can't attend class, what should I do?

It is OK, it happens! Students have the opportunity to cancel a lesson up to 8 hours before the scheduled time without loss of payment. So you can reschedule it for a convenient time, and the teacher will have the opportunity to

I don't have much free time, will I have time to study?

Learning can take place at your own pace. We will select a convenient schedule and at any time we will help you change the schedule, take a break or adjust the program.

How long is one lesson?

All classes last 1 hour.

Meet our team

Our teachers will tell you how to prepare for exams, help you cope with difficult tasks and win the Olympiad
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They will tell you about the pitfalls of exams and the most common mistakes, and explain how to avoid them

George Ionitsa

Founder &
Maths and Coding Coach

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