Find the 100th digit after the decimal point in the expansion of 1/7.
Division by strings of 9s; purely periodic fractions:
Compute by long division: 1 ÷ 9; 1 ÷ 99; 1 ÷ 9,999.
Prove the general rule:
1/(99…9) = 0,(00…01), where the denominator has n nines and the repeating block has length n with (n−1) zeros followed by 1.
Show that any purely periodic proper fraction equals a fraction whose numerator is the period and whose denominator is 10r − 1 (a number of r nines), where r is the period length.
Convert to decimal form:
23/99
1234/999,999
B. Repunits and divisibility
In the sequence 1, 11, 111, 1111, …, how many of the first 100 terms are divisible by 13?
If a repunit 11…11 (all digits 1) is divisible by 7, prove that it is also divisible by 11, 13, and 15,873.
In the sequence 1, 11, 111, 1111, …:
Prove there exist two terms whose difference is divisible by 196,673.
Deduce that there exists a repunit divisible by 196,673.
For any natural number a not divisible by 2 or 5, prove there exists a natural b such that the product ab is a repunit (written using only the digit 1).
If natural numbers a and m are coprime, prove there exists n such that an − 1 is divisible by m.
C. Cyclic digit moves and special base‑10 constructions
A k-digit multiple of 13 has its first digit moved to the end. For which k is the resulting number still a multiple of 13?
Example: 503,906 → 39,065 remains divisible by 13; 7,969 → 9,697 does not.
Find a six‑digit decimal number that becomes 5 times smaller when its first digit is moved to the end of the number.
The last digit of a (decimal) number is 2. If this digit is moved to the front, the number doubles. Find the smallest such number.
A five‑digit number is divisible by 41. Prove that any cyclic permutation of its digits is also divisible by 41.
Example: knowing that 93,767 is divisible by 41, conclude that 37,679 is divisible by 41.
D. Base‑16 (hexadecimal) curiosities
Does there exist a hexadecimal number that, when multiplied by 2, 3, 4, 5, and 6, yields numbers that are permutations of its digits (in base 16)?
Find a hexadecimal number that increases by an integer factor when its last digit is moved to the front (both operations understood in base 16).
Find quick answers to common questions about our lessons, pricing, scheduling, and how Exact Science can help your child excel.
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 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
What our students and parents say about learning with us
"Olympiad Maths Lessons helped me a lot to get the Gold medal in Junior Maths Challenge"
St. Paul's Student
"Thanks to the 'Data Science' and 'Coding in Python' lessons I got accepted to my dream university."
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