Mathematics Example 1.3

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Questions and Answers

What is the prime factorization of 72?

2^2 x 4 x 3

2^3 x 3 (correct)

3 x 24

2 x 3 x 12

If 9 divides 5544, what does it imply about 5544?

5544 is an even number

5544 is divisible by 3 (correct)

5544 must be prime

5544 is a multiple of 81

What theorem is applied in the solution to show that 72 divides the product of 5544 and 1176?

Fundamental Theorem of Arithmetic

The Principle of Mathematical Induction

Transitive Property of Equality

Theorem of Divisibility of Products (correct)

Which values are given to show the divisibility for 9 and 8?

99 and 168 respectively Signup and view all the answers

What conclusion can be drawn if both 9 divides 5544 and 8 divides 1176?

72 divides the product of 5544 and 1176 Signup and view all the answers

What are the positive divisors of 42?

1, 2, 3, 6, 7, 14, 21, 42 Signup and view all the answers

Using Theorem (1.2)(b), if 6 is a divisor of 42, which of the following is also true?

Any divisor of 6 is also a divisor of 42. Signup and view all the answers

What is the value of the tau function τ(42)?

8 Signup and view all the answers

Which of the following numbers is NOT a divisor of 42?

8 Signup and view all the answers

How can negative divisors of a number be determined?

By adding a negative sign in front of the positive divisors. Signup and view all the answers

What can be inferred if a divides 1?

a is equal to ±1. Signup and view all the answers

What does it imply if a divides b and b divides a?

a = ±b. Signup and view all the answers

Which of the following statements about divisibility is true?

If a divides b and b divides c, then a divides c. Signup and view all the answers

How is the concept of integers utilized in the proof?

To apply properties like associativity in calculations. Signup and view all the answers

What does the symbol ⇨ signify in the mathematical proof?

An implication. Signup and view all the answers

What are the positive divisors of 21?

1, 3, 7, 21 Signup and view all the answers

Which of the following numbers is NOT a divisor of 100?

3 Signup and view all the answers

According to the properties of divisors, if a | b and b | c, what can be determined?

a | c Signup and view all the answers

Which of the following statements about divisors is true?

If a | b and b | a, then a = b Signup and view all the answers

What are the divisors of -21?

1, 3, 7, 21, -1, -3, -7, -21 Signup and view all the answers

What does it mean when we say that an integer a divides another integer b?

There exists an integer m such that a × m = b. Signup and view all the answers

How do we denote that integer a does not divide integer b?

a ∤ b Signup and view all the answers

If 5 is a divisor of 100, what can also be inferred?

100 is a multiple of 5. Signup and view all the answers

Which of the following statements is true regarding divisors?

Divisors are interchangeable terms with factors. Signup and view all the answers

What is a consequence of the definition of divisibility for an integer 21?

21 is both a multiple and a divisor of itself. Signup and view all the answers

Study Notes

Example 1.3

Given that 99 | 5544 and 168 | 1176, show, without using a calculator, 72 | (5544 x 1176).

Solution

Since 72 = 9 x 8, so we need to show (9 x 8) | (5544 x 1176).
Clearly 9 | 99 and we are given 99 | 5544, therefore 9 | 5544.
Similarly 8 | 168, and we are given 168 | 1176, therefore 8 | 1176.
Applying Theorem (1.2)(c): If a | b and c | d then (a x c) | (b x d).
To 9 | 5544 and 8 | 1176 gives (9 x 8) | (5544 x 1176), which implies 72 | (5544 x 1176).

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Description

This quiz demonstrates a problem involving divisibility in mathematics. It shows how to apply theorems related to factors and divisibility to prove that one number divides another through given relationships. Engage with the concepts of factorization and divisibility calculations.