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Prime Time
Whole Number Arithmetic
A little vocabulary
Factors are one of two or more
whole numbers that are multiplied
together to get a Product
5 x 7 = 35
Five and seven are factors, and
thirty-five is a product
There are 2 kinds of
numbers…
Prime Numbers
These numbers have only two factors… One and
themselves
7=7x1
7: 1,7
Composite Numbers
Have at least three factors
10 = 5 x 2 and 10 x 1
10: 1,2,5,10
12 = 6 x 2 and 4 x 3 and 12 x 1
12: 1,2,3,4,6,12
Factor Pairs
24: 1, 2, 3, 4, 6, 8, 12, 24
All factors have pairs
Finding All of the Factors
You have to be systematic
Start with one and count up
Stop when you find two consecutive factors
whose product is the number you are
working on.
60: 1, 2, 3, 4, 5, 6, 10
6 x 10 = 60 so this is the central factor pair
Now just match your other factors!
Square Numbers
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
3 x 3 = 9. This statement could be modeled like this:
Square Numbers
16: 1, 2, 4, 8, 16
Because we don’t repeat factors in our lists, square
numbers have an odd number of factors
4 x 4 is the middle pair
The square root of a square number is a whole
number!
Numbers between square numbers have square roots
between the squares…
Square Roots
The numbers between 25 and 36 have square roots
between 5 and 6 (non whole numbers)
Examine Factor Pair Mountain (website)
Knowing our square roots, tells us when to stop
looking for factors.
27’s square root is between 5 and 6 so I only need to
count to 5 to find all of the numbers up FPM
27: 1, 3 – and we’re ready to climb down!
Divisibility
1: All numbers have 1 as a factor
2: All even numbers (ending in 0, 2, 4, 6, 8)
have 2 as a factor
3: If the sum of the digits is divisible by 3 so is
the number itself
81 -> 8 + 1 = 9, which is divisible by 3, so 81 is
too!
372 -> 3 + 7 + 2 = 12, so 3 is a factor
109 -> 1 + 0 + 9 = 10, so 3 is not a factor
Divisibility
5: Numbers that end in 5 or 0 have five as a factor
6: Numbers that have 2 & 3 as factors have 6
9: Same trick as for 3!
162 -> 1 + 6 + 2 = 9
162 = 9 x 18
So what about the rest of the numbers?
This is where it gets interesting…
Stretching Numbers
How do we know if 8 goes into 140?
Our math facts don’t go up that high!
Long division?
Pick a number close to 140 that you know 8 goes
into… how about 80.
Look at the difference (60)
If 8 goes into the difference, it goes into the
number
Stretching Numbers
See if 8 goes into 224
You might use 8 x 3 to come up with 8 x 30
Since we know 8 goes into 240 all we have to
do is check the difference – in this case 16. This
means that 8 is a factor of 224. (8 x 28 = 224)
Let’s practice!
Primes
The Fundamental Theorem of Arithmetic
All numbers are the unique product of prime numbers
I think of this as a unique fingerprint for each
number, or perhaps a number’s true name (Eragon)
20: 2 x 2 x 5
23: 1 x 23 (prime)
21: 3 x 7
24: 2 x 2 x 2 x 3
22: 2 x 11
25: 5 x 5
Factor Trees
36
4
2
9
2
3
36: 2 x 2 x 3 x 3
3
Factor Trees
42
6
2
7
3
7
42 = 2 x 3 x 7
Factor Trees
You can start any way you want, but the
end result will always be the same – the
order of the factors does not matter
Examine different factor trees for 100
Practice!
LCM & GCF
Knowing the prime factorization of numbers allows
us to see what they have in common.
Greatest Common Factor (GCF)
ALL of the prime factors that any two (or more) numbers
have in common.
INTERSECTION
Least Common Multiple (LCM)
ALL of the prime factors (without duplications)
UNION
LCM & GCF
24
36
2
3
Going Forward
Computation is probably always going to be a part of
your child’s math education – have them master the
facts now.
Do mental math at every opportunity with them
Not just the facts
Estimation
Divisibility tests
Large numbers
Next Units
Bits and Pieces (CMP)
Fractions – representations and computation
Potential parent night?
Division (Additional Materials)
Long Division – standard algorithm
Estimation
Mental Math
Thanks
There will be more group practice if time permits