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Part B:

This session introduces some topics related to number theory. Number theory allows us to consider why mathematics works the way it does. You will work with “Alpha math” problems to explore relationships among numbers, which is an important part of thinking about mathematics. You’ll look at divisibility tests and why they work, and then move on to examine factors.

**LEARNING OBJECTIVES**

In this session, you will do the following:

• Use number theory to build your problem-solving skills

• Use “Alpha math” problems to deepen your understanding of relationships among numbers and build your problem-solving skills

• Find and use divisibility tests for 2, 3, 4, 5, 6, 8, 9, 10, and 11 and understand why these tests work

• Begin to explore and understand factors

**Previously Introduced:**

base

The base of a number system is the number representing the value of each place in a representation. For example, “base ten” tells us that each digit in a number is some value of 10. In base ten, the number 1,234 represents four different values of 10: (1 • 10^{3}) + (2 • 10^{2}) + (3 • 10^{1}) + (4 • 10^{0}). Meanwhile, 1,234 in base five represents (1 • 5^{3}) + (2 • 5^{2}) + (3 • 5^{1}) + (4 • 5^{0}), and so on. These representations may appear identical, but if you perform the calculations, you’ll see that 1,234 in base ten is a different number from 1,234 in base five.

**New in This Session:**

divisibility test

A divisibility test is a rule that determines whether a given number is divisible by a set factor. For example, we can use a divisibility test to determine if a large number like 23,456 is or is not divisible by 2, by 3, or by 5. Some divisibility tests involve the last digits of a number, while others involve the sum of the digits.

**factor**

A factor of a number is a counting number that divides evenly into that number. For example, 3 is a factor of 15, since 3 divides evenly into 15 (five times). Four is not a factor of 15, but it is a factor of 16.

**factor tree**

A factor tree can be used to factor a number into prime factors. To create a factor tree, start with the smallest prime factor of the given number and then split the number into factors. With 30, the smallest prime factor is 2, so 30 = 2 • 15. Then factor 15 into prime numbers: 30 = 2 • 15 and 15 = 3 • 5. So, 30 = 2 • 3 • 5, which is its prime factorization.

**figurate number**

A figurate number is a number of dots which form a geometric shape. If you make a square with 5 dots on a side, there will be 25 dots; this makes the number 25 a square number. If you make a triangle with 4 dots on a side, there will be 10 dots; this makes 10 a triangular number. Figurate numbers can be formed from pentagons, hexagons, cubes, pyramids, and other geometric shapes.

**prime number**

A counting number is a prime number if it has exactly two factors: 1 and the number itself. For example, 17 is prime, 16 is not prime, and 1 itself is not prime, since it has only one factor.

**relatively prime numbers**

Two or more numbers are relatively prime numbers if their greatest common factor is 1. For example, 4 and 9 are not prime numbers, but they are relatively prime because their greatest common factor is 1.