This lesson was based on material from Dr. George Lenchner's book, Creative Problem Solving in School Mathematics.
Divisibility
In the decimal system, the easiest factors to check for are 2, 5, and 10, because you can tell just by looking at the ones digit.
Here are some problems involving factors of 5:
- My calendar numbers the days of the year. For example, today is day 316 of 2013. This year (like most) has 365 days in it. For how many days is that day's number both divisible by 5 and has the digit 5 in it?
- How many even days are divisible by 5?
- How many contain the digit 5 but are not divisible by 5?
Parity
Last week I mentioned that in English (and many other languages) we have special words for things to do with two: for example, both, between, pair, and even. We don't have a special word for numbers that are or aren't multiples of 3, or 5, but we do for 2.
What happens when you add odd and even numbers? What is the parity of the result of adding these kinds of numbers?
- Two even numbers?
- Two odd numbers?
- One even number and one odd number?
- Five even numbers?
- Five odd numbers?
- An even number of even numbers?
- An odd number of even numbers?
- An even number of odd numbers?
- An odd number of odd numbers?
- One hundred and nine odd numbers and three even numbers?
Sums and Differences
One of the harder numbers to check as a factor is 7.
With divisibility, you can always resort to multiplication or division, but here is a trick that is sometimes faster.
Suppose you have two numbers, a and b, and a is bigger than b. Suppose you know that a is divisible by 7, and b is divisible by 7.
Is a + b divisible by 7?
What about a - b?
Here's another way of looking at it:
Suppose you have some number, like 329, and you're not sure if it is divisible by 7. You can take away as many multiples of 7 as you like, and the number left will have the same remainder when divided by 7 as the original.
So you might notice that 280 is a multiple of 7 and take that away. Now you have 49 left. Since 49 is a multiple of 7, you know 329 must be, too.
Are these divisible by 7?
- 56+28
- 35+65
- 210-63
- 770-350
Watch out for this gotcha:
Is 26+16 divisible by 7?
Taking away multiples of 7 doesn't change the remainder, but taking away non-multiples of 7 does!
Which of these makes a whole number of weeks?
- 99 days
- 153 days
- 677 days
