IME2076 - What has elementary number theory got to do with school arithmetic?

  • Arithmetic is the very first topic covered in most Secondary One mathematics. Students learn about factors, primes, prime factorisation, highest common factor and lowest common multiple. One application of prime factorisation is finding square roots and cube roots of whole numbers. For example, to find the square root of 2025, we express 2025 as product of primes 34 x 52, from which we obtain the answer 32 x 5. On the other hand, the square root of 200 is not a whole number because 200 = 23 x 52 and the power of 2, which is 3, is odd. In contrast, we cannot say that the square root of 324 = 27 x 12 is not a whole number because the power of 27 is odd. Of course, we argue that we cannot make this conclusion because 27 x 12 is not a prime factorisation of 324. So what is so special about prime numbers? Why do we have to write a whole number N in prime factorisation form to determine conclusively whether the square root of N is a whole number or not? In this course, we shall discuss the mathematical principle behind this, and also explain why the algorithm for finding HCF and LCM using prime factorisation works.

    We shall also touch on representation of real numbers in decimals. What is the difference between rational and irrational numbers written in decimals? How do we know whether a fraction m/n written in decimal is terminating or non-terminating (2/5 = 0.4 is terminating, 2/3 = 0.666… is non-terminating)? These questions can be answered using elementary number theory.

    We also discuss some real-life applications of number theory; for example, how is the official reference of NRIC No., or the ISBN of books, obtained? We shall share some interesting mathematical puzzles involving number theory. For example, can you get four gallons of water using five and three gallon jugs with no markings? (In the movie “Die Hard 3”, John McClain (played by Bruce Willis) and Zenus (played by Samuel L. Jackson) have to solve this puzzle posed by the villain Peter Krieg (played by Jeremy Irons) in order to defuse a bomb. Had the villain asked them to get four gallons using three and six gallons jugs, would they be able to solve it?)

    Learning Dimension: Deepening Knowledge and Practice in Academic Curriculum

  • To understand the mathematical principles behind some topics covered in secondary school arithmetic, thereby gaining competence and confidence in teaching these topics.

  • Factors, division algorithm, prime numbers, the Fundamental Theorem of Arithmetic and its consequences, highest common factor and lowest common multiple, the Euclidean algorithm, linear Diophantine equations, some real-life applications of number theory, some puzzles involving number theory.

  • Secondary School Mathematics Teachers

    *This course is also open for customization to schools & clusters

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  • Duration: 9 Hours

    Note: NIE reserves the right to change or cancel the course due to unforeseen circumstances.

    • MOE Participants - $97.30 (incl. 7% GST & $1 Copyright Fee).
    • Non MOE Participants - $723.25 (incl. 7% GST & $1 Copyright Fee)
  • Please read the instructions below carefully before completing the application form.   

    Upon confirmation, payment is required for all courses. Payment should only be made after the course placement letter has been sent to participants to confirm the application. Cancellations/withdrawals are not allowed. However, a replacement may be sent to attend the same course. 

    Payment through GIRO/IFAAS

    NIE will proceed to invoice the respective schools, and they can  make payment through GIRO or IFAAS. The amount payable should be credited into the NIE bank account number 518-041272-001.

    Payment by Cheque

    Please submit the cheque together with the application form. Cheque should be made payable to “National Institute of Education” and participant(s)’ name and course(s) clearly indicated on the reverse of the cheque.

    Please mail to: 
    National Institute of Education
    Office of Graduate Studies and Professional Learning
    1 Nanyang Walk, Singapore 637616
    Attn: Ms Nur Shafiqah


    NIE will acknowledge receipt of the application form through email within one week. The course placement letter will be emailed to applicants who have been successfully placed in a course and upon course confirmation.


    Please note that NIE reserves the right to replace an instructor for a course if the lecturer is unable to teach the course due to unforeseen circumstances. No refund of the course fee will be made to participants.

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    Applicants will be notified of the results of their applications via e-mail.