Mathematics
Problem of The Day
2024
111

111

For every natural number n, prove that the expression n5 - 5n3 + 4n is always divisible by 120.

Answer
SOLUTIOn

We begin by factoring the expression n5 - 5n3 + 4n as n(n2−1)(n2−4)=(n−2)(n−1)n(n+1)(n+2).

Notice that among five consecutive integers, one is divisible by 5, at least one is divisible by 3, and two are consecutive even numbers, one of which is divisible by 2 and the other by 4.

Therefore, the expression is divisible by 2×4×3×5=1202×4×3×5=120.

‍

Previous
Next

Our Classes

Mathematics
Programming
Chess
Geometry
Curriculum Mathematics
Mathematics
8+

Curriculum Mathematics

A complete maths course for exam success. Strengthen core skills, tackle advanced topics, and prepare for school, GCSE, or A-Level maths exams with confidence.

Olympiad Mathematics
Mathematics
8+

Olympiad Mathematics

Join our Olympiad Maths training for ambitious students. Build skills in key topics like algebra, geometry, and combinatorics while practising with real past papers and simulated challenges.

Introduction to Python with Turtle
Programming
9+

Introduction to Python with Turtle

This course guides you through the fundamentals of Python programming using an interactive Python library known as Turtle.

Coding in Python
Programming
8+

Coding in Python

Start coding with Python – a simple yet powerful language for beginners. Learn to write programs, solve problems, and automate tasks through hands-on lessons.

Data Science
Programming
16+

Data Science

This course is designed for those with little or no prior programming or data science experience, but who are eager to learn how to use Python to analyze and visualize data.

Chess
Chess
7+

Chess

Supercharge your potential by studying chess alongside maths and programming. Boost strategic thinking, problem-solving, and concentration. Unleash your competitive edge and unlock endless possibilities in these fields.

Geometry
Geometry
12+

Geometry

This course encompasses a range of Geometry topics such as coordinate and spatial geometry, introductory trigonometry, angles, parallel lines, congruent and similar triangles, polygons, circles, the Pythagorean Theorem, and more. Emphasis will be placed on reinforcing Algebra skills and enhancing critical thinking through problem-solving in both mathematical and real-world contexts.

FAQ

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.

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."
Michael
Data Science Student
Warwick University
"Great courses, which thoroughly explained topics beyond the capability of the GCSE answer sheet. Thanks so much."
Ivan
GCSE Student in Dubai
"Financial Mathematics! Best course to understand Python and Mathematics behind Finance!"
Gleb
VC Investor
"We got silver in PMC! Thanks George!"
Mum of St. Paul's Student
Prepare for the Primary Maths Challenge
"My daughter took a batch of 10 classes with George to understand Python with Turtle. I found George extremely knowledgeable and accessible."
Dad of Latymer School Student
Python with Turtle

Contact us

Email

george@exactscience.co
‍

Phone

Mon-Sun from 8am to 9pm
+44 77 07 547 144

Location

36 Lupus St, Pimlico, London SW1V 3AT

...or ask us anything

Ask about our courses and offerings, and we will help you choose what works best for you.

Thank you! Your submission has been received!
Oops! Something went wrong while submitting the form.