Divisibility by 3 and a Contest

Sums and differences again: remainders

Last time we talked about using sums and differences to tell if a number is divisible by 7.

For example, 781 is not divisible by 7: $781=770+11$, and we know that 770 is divisible by 7 and that 11 is not.

A similar kind of logic works with remainders.

• What is the remainder (R) when you divide 465 by 7?

  $465=420+45$.

  $45÷7$ → R3.

• What is the remainder if you divide (13 + 283) by 7?

• Method 1: $13+283=296=280+16$. $16÷7$ → R2
• Method 2: $13÷7$ → R6; $283÷7$ → R3; $6+3=9$; $9÷7$ → R2, so $(13+283)÷7$ → R2

The divisibility by 3 "trick":

If the digits in a number sum to a multiple of 3, then the number is divisible by 3.

How do we know this works?

First, see if there are any obvious exceptions: does it work for the first few multiples of 3 you can think of? How about the first few non-multiples of 3?

Okay, so you may have persuaded yourself that there is something to it, but does this prove it always works?

Of course not!

Here's the reason it works:

$10÷3$ → R1

So if you have, for example 54, $54=50+4$. You know that 50 will have a remainder of 1 when divided by 3, and that will happen 5 times. As far as remainders are concerned, this is just like adding 5.

We had to stop here to start...

Our first contest

which was really fun!

We'll go over the solutions, not next week, but the week after.

More on Divisibility

Last week we took a brief look at factors and primes, and today we talked more about divisibility.

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

First Day!

Today at Math Olympiad we talked about 3 things:

• What does it mean to be good at math?
• What is Math Olympiad?
• Factors / Divisibility / Primes

Here is a quiz about being good at math:

Q. Is being good at math something you just naturally are or aren't?

A. Some people find math easier than other people, but the way you get good at math is by practising.

Q. If I make a lot or mistakes, or don't "get it" right away, does that mean I'm bad at math?

A. No! Learning a new math concept or technique takes practice.  Even professional mathematicians make mistakes sometimes and take time to develop their skills.

Q. I am a girl.  Is it weird for me to like math?  Am I naturally not as good at math because I'm a girl?

A. Absolutely not.  It's normal for girls to enjoy and excel at math.

If you believe you are good or bad at something, that can be a self-fulfilling prophecy, because if you think you are good at it, you will like it and keep trying, but if you think you are bad at it, you will stop trying.  Don't fall for the trap of believing you are bad at math!


What is Math Olympiad?

Math Olympiad is a problem solving program.  We meet once a week and practise solving fun math problems.  Once a month we try our hands at contests.

The contests always consist of 5 problems.  They have a single answer, and regardless of how well or poorly you approached the problem, your score on that problem is just 0 or 1, depending on whether you got it wrong or right.

Q. If I get only 1 or 2, or even no questions right, does that mean I'm bad at math?

A. Getting the right answer is fun and satisfying, but getting it wrong is okay, too.  The problems are designed to be tough!  Instead of comparing yourself to others, try to figure out how you can improve.

A Sample Problem

We looked an old contest, and tried the first problem.  The first problem often looks like a lot of arithmetic, but turns out to be easy if you rearrange it.

For example, we looked at a problem that asked:

What is the value of:

55 - 44 +11 - 22 + 33 - 33 + 22 - 11 + 44 - 55.

See if you can spot the trick that turns this into a very easy problem!
If you don't see it, don't worry.  There are usually many ways to solve a problem, and the best one to use is the one that makes most sense to you.

Factoring and Primes

A lot of Math Olympiad problems are based on knowing how to find factors.

A factor is just a fancy word for a natural number that divides evenly into another.
(A natural number is a counting number: 1, 2, 3, 4, ....; not a negative, decimal, or fraction)

So, for example, the factors of 6 are 1, 2, 3, and 6.

A natural number that has exactly 2 factors (itself and 1) is called prime.

For example, 2, 3, and 17 are prime.  15 is not prime, because it has 1, 3, 5 and 15 as factors.

1 is not prime.  It has only 1 factor.

How can you tell if a number is prime?  We checked 51 to see if it was prime by seeing if any number was a factor, starting with 2.

Checking all the numbers to see if they are factors can take a lot of time, especially if the number is big, or if we have a lot of numbers to check.

We made a Sieve of Eratosthenes to generate primes below 100.  Wikipedia has a nice animation of exactly what we did by hand.

Next time


Next time we will do more with factors, and talk about how to recognise when a number is divisible by other numbers.

Please join us!