Determine whether each number
is prime or composite.
A prime number is a natural number or counting number
greater than one that has only two factors,
one and itself.
A prime number is also
only divisible by one and itself.
And here's a list of the first several prime numbers.
A composite number is a natural number or counting number
greater than one that has more than two factors.
If a number is not prime,
it is composite.
And for review,
if a and b are counting numbers,
and a times b equals n,
then a and b are factors of n.
A and b also divide evenly into n.
The first number we are given is 15.
Because one times 15 equals 15,
one and 15 are factors of 15.
To determine additional factors of 15,
we need to find two additional counting numbers
that give a product of 15.
And because three times five equals 15,
three and five are also factors of 15.
Which means one, three, five, and 15
are factors of 15.
15 has more than two factors.
15 is composite.
It is important to recognize
we don't have to find all the factors of a given number
to determine whether it's prime or composite.
As soon as we find more than two factors,
a given number is composite.
The next number is 17.
Because one times 17 equals 17,
one and 17 are factors of 17.
To determine additional factors of 17,
we need to find two additional counting numbers
that give a product of 17.
Because this is not possible,
17 only has two factors,
one and itself,
which means 17 is prime.
The next number is 37.
We know one times 37 equals 37,
which means one and 37 are factors of 37.
To determine additional factors of 37,
we need to find two additional counting numbers
that give a product of 37.
Well six times six is 36, not 37.
It's not possible
to find two additional counting numbers
that give a product of 37,
which means 37 only has two factors,
one and itself.
37 is prime.
Next, we have 81.
One times 81 equals 81.
We should also be able to recognize that nine times nine
is equal to 81,
so now we know that one, nine, and 81 are factors of 81.
81 does have additional factors,
but we don't have to find all the factors
because we know 81 has more than two factors,
81 is composite.
Remember the divisibility rule for three,
if the sum of the digits is divisible by three,
so is the number.
Eight plus one is equal to nine.
Because nine is divisible by three,
it also divides 81.
Three would also be a factor of 81.
The last number is 127.
Well of course you know one times 127 equals 127.
When the numbers get larger,
it becomes much more difficult to just find
two additional counting numbers
that give a product of the given number,
so let's end by going through a step by step process
to determine whether a number is prime or composite.
The steps are outlined below.
Where we test each of the primes in order
to see if it is a factor of a number.
To do this, we divide the given number by the prime number.
If it has a remainder, it's not a factor.
If the remainder is zero, it is a factor.
We start with two and stop when the quotient
is smaller than the divisor
or when a prime factor is found.
If the number has a prime factor,
then it is a composite number.
If it has no prime factors,
then the number is prime.
So we begin with 127.
And we divide by the first prime which is two,
but we should recognize that two is not going
to divide 127
because 127 is odd.
But to go through the process,
there are 63 twos in 127.
63 times two is equal to 126,
we subtract, the remainder is one.
Again because we have a remainder,
two is not a factor of 127.
Notice here the quotient of 63
is not smaller than the divisor of two,
which means we need to divide by the next prime
which is three.
So now we have 127 divided by three.
There are four threes in 12,
subtract, bring down the seven.
There are two threes in seven,
two times three is six, we subtract.
Again we have a remainder
which means three is not a factor of 127.
But 42 is not less than three, we divide by the next prime
which is five.
There are two fives in 12,
two times five is 10, we subtract,
bring down the seven.
There are five fives in 27,
five times five is 25, we subtract,
again we have a remainder
which means five is not a factor of 127,
but because 25 is not less than five,
we divide by the next prime
which means we now divide by seven.
There's one seven in 12, subtract,
difference is five,
bring down the seven.
There are eight sevens in 57.
Eight times seven is 56, we subtract,
again we have a remainder.
Seven is not a factor of 127.
Because 18 is not less than seven,
we divide by the next prime which is 11.
There is one 11 in 12,
one times 11 is 11, subtract.
The difference is one.
Bring down the seven.
There's one 11 in 17,
one times 11 is 11.
Again we have a remainder, the difference is six.
11 is not less than 11,
which means we need to divide by the next prime which is 13.
And this will be the last prime we need to divide by.
We need to determine how many 13s in 127,
which I believe is nine.
Nine times three is 27, form an exchange with the two,
nine times one is nine plus two is 11.
We subtract, have a remainder of 10,
which is less than 13 so the nine is correct.
But because we have a remainder,
13 is not a factor of 127,
and now because nine is less than 13,
we can stop.
We now know 127 is prime
because it only has two factors,
one and itself.
So this process can be time consuming,
but when we have a large number,
this is one way to be sure
whether a number is prime or composite.
Again 127 is prime.
I hope you found this helpful.