Proof: If the sum of the digits of a number is divisible by 3 then the number is divisible by 3

Given a number with digits of the form   If the sum of these digits is equal to a number that is divisible by 3 then the number is divisible by 3.  Proof follows Starting with      with  which must be 0.3,6,9 then      is said to be divisible by Read more…

NonTransitive Dice

Research Links Google: non transitive dice Wikipedia: Non transitive dice Wikipedia: Miwin's dice Non transitive Grime dice £11.93 Consider the following set of dice. Die A has sides 2, 2, 4, 4, 9, 9. Die B has sides 1, 1, 6, 6, 8, 8. Die C has sides 3, 3, 5, 5, 7, 7. The probability that A rolls a higher number Read more…

New Zealand Mathematical Olympiad 2019 Question 5

We present a solution to round 1 question 5 from the 2019 New Zealand Mathematical Olympiad. This problem involves describing all values of a quartic expression which is a perfect square.

Integer Square 1 to 100

Walking the beira mar here in Sao Jose, SC. Keeping my mind busy during the fake virus scare / Bank Robbery.   This entire entry is about what I am learning from calculating the squares of the numbers 0 through 100 mentally as I walk the beira mar.  First thing you Read more…

British Mathematics Olympiad 1993 Round 1 Question 1

Notes Used extended Euclidean algorithm Uses and demonstrates factor substitution method well

Diophantine Equations

Research Links Google: diophantine equation pythagoras Chapter 1: DIOPHANTINE ANALYSIS – UnKnown Book Diophantine Equations – Chapter 5 –  Diophantine Analysis monograph by Robert Daniel Carmichael Pythagorean Triples Solution The square of the odd number         is      Hence the sum of two odd squares is divisible by 2 but Read more…

Solution to Question A1 from the 2018 Putnam – Modulus Arithmetic

A good example of usage of modulus arithmetic.