Multiply an odd number by the two numbers either side of it. Prove that the final product is divisible by
Mattia is thinking of a big positive integer. He tells you what this number to the power of
You might want to know what day of the week your birthday is this year. Mathematician John Conway invented an algorithm called the ‘Doomsday Rule’ to determine which day of the week a particular date falls on. It works by finding the ‘anchor day’ for the year that you’re working in. For
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Then find the nearest one of these dates to the date that you’re looking for and find remainders.
For example,
What day of the week will
That is, we count A,B,C vs. D,E,F as the same splitting as F,D,E vs A,C,B.
Below is a regular octagon. Given that its side length is
In the diagram, all the small squares are of the same size. What fraction of the large square is shaded?
The letters
Find
A remainder is the number that is “left over" from division. Even if a number is not divisible by another number fully, we can still divide, but leaving a remainder. The remainder is less than the number we’re dividing by. For example, a remainder of
More generally, given any integer
The general rule is that the remainder of a sum, difference or a product of two remainders is equal to the remainder of a sum, difference or a product of the original numbers. What that means is if we want to find a remainder of a product of two numbers, we need to look at the individual remainders, multiply them, and then take a remainder.
For example,
Here is a useful notation when discussing problems involving remainders and divisibility although it is not necessary for this problem sheet. Take two integers
Using this new notation, we can easily express the rules for remainders. Let
We make one last observation, which shows the utility of remainder when discussing divisibility. Saying that a number
Let’s have a look at some examples with remainders:
Picasso colours every point on the circumference of a circle red or blue. Is he guaranteed to create an equilateral triangle all of whose vertices are the same colour?