CTK Insights

For a prime , two integers are both divisible by with the probability , because this only happens when the two integers have the residue 0 (one out of available residues) modulo . Two integers are mutually prime if they have no common nontrivial factors, prime facors in particular. Assuming divisibility by one prime is [...]

It's hard to overestimate the influence Alcuin of York (c. 732-804) had on Western civilization. He also left the earliest known European collection of puzzles, Propositiones alcuini doctoris caroli magni imperatoris ad acuendos juvenes - Propositions by Alcuin Teacher to the Great Emperor Charles to Sharpen up the Young. The collection consists of 53 problems [...]

Parabola has an easily verifiable property. The segment joining points and crosses -axis in point . The equation of the segment is , from which . This may be a curious fact in its own right. What does it say? Taken at a face value, it simply shows a way to obtain the product of [...]

According to [P. Beckmann, p. 12], by 2000 BC, the ancient Babilonians already new that π is close to 25/8 (≈ 3.125), while the Egyptians estimated it as 4(8/9)² (≈ 3.1605). Beckmann, as many others, do not conceal his astonishment that in the Bible the apparent estimate is a simplistic 3. This is based on [...]

Every book carries an ISBN or EAN-13 or both numbers. ISBN or EAN-13 are two ways to encode information that would allow to identify a book on the market. ISBN contains 10 digits EAN-13 thirteen. All but the last digit encode country, manufacturer (publisher), and product (book) information. The last digit serves as a safeguard [...]

The PEMDAS acronym is supposed to help students memorize the order of operations. I consider introducing and using it an education outrage. Let me explain.

The setup Write a sequence of integers from 1 through some number n, say, 20-30. Do that with ink. Draw a square around each number. (Draw squares with a pencil; they need to be erasable.) The activity The activity proceeds in a sequence of steps. On step #i consider all multiples of i. If a [...]