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(Answer ID # 0830384)
Number Theory
Complete.
1. Use the digits 6, 5, and 7 to create a 3- 2. Identify a number that is divisible by
digit number that is divisible by 2, 3, 4,
12, 2, 6, 16, and 3, which is not
6, 8, and 9. Describe how you found the
divisible by 7, 18, 17, 19, and 13.
number.
Describe how you found the number.
3. Identify two numbers whose GCF is 16 4. Prime numbers tend to get less and less
and whose LCM is 96. Describe how
frequent as they get larger in the
you found the number.
positive direction. Why do you suppose
this is?
5. Multiplication is sometimes defined as
"repeated addition". Give an example
of how a multiplication problem is an
easier way of writing a repeated
addition.
6. Numbers that end in zero are
considered even because they are
definitely divisible by two. However,
the number 0 is neither even nor odd.
Think about this, and give an
explanation of why this must be true.
7. If the sum of the digits of a number is 8. The sum of two opposites is very fast to
divisible by three, then the number
calculate. Why?
itself is divisible by three. If the sum of
the digits of a number is divisible by
nine, then the number itself is divisible
by nine. For example 36 is divisible by
both 9 and 3 because 6 + 3 = 9, and 9 is
divisible by both 3 and 9. Are all
numbers that are divisible by 3 also
divisible by 9? Why or why not?
9. A strange signal sequence has been
10. The divisibility rules are helpful in
detected by radio astronomers. They
math because they allow us to analyze
have recorded a signal consisting of an
division problems to arrive at a quick
alternating series of ones and negative
solution. The number 6 is divisible by
ones {1, -1, 1, -1, 1, -1,…}. Scientists
both 2 and 3. How can you tell if a
want to understand the meaning of this
number is divisible by 6? Show by
signal. They look for patterns. The first
example a number that is divisible by
term of the sequence is 1. What will
2, 3, and 6.
the 990th term be? How do you know?
Describe a rule to figure out the answer
for the xth number.
11. Ms. Floop has a saying in her math
class that she uses to help students
remember a certain fact about
numbers. She mentions it every time
there is a discussion about number
systems. Her saying is “Counting is
Natural!” What do you think she
means by this, and what numbers is
she talking about?
12. There are two commutative laws in
introductory math: the commutative
law of addition and the commutative
law of multiplication. What do these
two laws have in common? Hint:
Write down the equations for the
commutative laws of addition and
multiplication.
Answer Key 0830384
576
96
32 and 48
Because the larger a number is the greater the probability that we will be able to find
divisors for the number and, thus, the probability of finding prime numbers decreases.
5 Adding a number to itself n times is equivalent to multiplying that number n+1 times.
6 Dividing a number by 0 (n/0) is an undefined operation and has no meaningful result.
Thus 0 is not divisible by 2 or any other number so it is impossible to determine if 0 is
even or odd.
7 No. The easiest way to show this is by considering the number 6. Six is clearly
divisible by 3 but not by 9. There are many other examples. Thirty-three is divisible
by 3 because 3 + 3 = 6, which is divisible by 3. But 33 is not divisible by 9.
8 Because summing two opposites always has the same result, which is easy to
memorize. The answer is 0.
9 -1. The sequence repeats itself after two terms (1, -1). Every even term (the second,
fourth, sixth, etc.) always is a -1. Every odd term is a 1.
10 If a number is divisible by 6, it is always also going to be divisible by 2 and 3 which
are factors of 6. So to tell if a number is divisible by 6, check it for divisibility by both
2 AND 3. Eg. 36 ÷ 6 = 6, 36 ÷ 2 = 18, 36 ÷ 3 = 9. There are many other examples.
11 She uses the saying as a reminder that counting numbers and natural numbers are two
names for the same set of numbers, {1, 2, 3,}.
12 The most striking similarity is that they both have to do with the way numbers are
ordered in an expression. Sometimes the order of numbers matters, and sometimes it
doesn’t. In addition you always get the same result when adding two numbers, a and
b, no matter if you add a to b, or add b to a.
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