1

A four-digit number, n, is written as ‘ABCD’ where A, B, C and D are all different odd digits. It is divisible by each of A, B,C and D.

Find all the possible numbers for n.

Answer

All the possible numbers for n are 1395, 1935, 3195, 3915, 9135, 9315.

SOLUTIOn

‘ABCD’ is a number composed of four out of five digits 1, 3, 5, 7, 9. Consider two cases:Case 1. 9 is not one of the digits A,B, C, D.

Number ‘ABCD’ is a multiple of 3, so A + B + C + D is also a multiple of 3. But A + B + C + D = 1 + 3 + 5 + 7 = 16, which is not a multipleof 3. This is a contradiction, so there are no solutions in this case.

Case 2. 9 is one of the digits A, B, C, D.

Number ‘ABCD’ is a multiple of 9, so A + B + C + D is also a multiple of 9.

Let x be the digit not used in ‘ABCD’.Then A + B + C + D = 1 + 3 + 5 + 7 + 9 − x = 25 − x

and this must be a multiple of 9. Hence x = 7.

This means 1, 3, 5, 9 are the digits used, so ‘ABCD’ is a multiple of 5. Hence D = 5. We can rearrange 1, 3, 9 as A, B, C in six different ways.

All the possible numbers for n are 1395, 1935, 3195, 3915, 9135,9315. All of them are divisible by 1, 3 and 9 (as their sum of digits is always 18), and also by 5 (as they all end in a 5).

Subscribe to receive updates!

Be up-to-date with our recent updates, new problems and answers!

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

Our Courses

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.
Hear from some of our amazing students who already achieved incredible results with us!
"Olympiad Maths Lessons helped me a lot to get the Gold medal in Junior Maths Challenge"
St. Paul's Student

Contact us

...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.