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?

<p>99 and 168 respectively (D)</p> Signup and view all the answers

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

<p>72 divides the product of 5544 and 1176 (D)</p> Signup and view all the answers

What are the positive divisors of 42?

<p>1, 2, 3, 6, 7, 14, 21, 42 (B)</p> 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?

<p>Any divisor of 6 is also a divisor of 42. (C)</p> Signup and view all the answers

What is the value of the tau function Ï„(42)?

<p>8 (A)</p> Signup and view all the answers

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

<p>8 (A)</p> Signup and view all the answers

How can negative divisors of a number be determined?

<p>By adding a negative sign in front of the positive divisors. (A)</p> Signup and view all the answers

What can be inferred if a divides 1?

<p>a is equal to ±1. (C)</p> Signup and view all the answers

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

<p>a = ±b. (C)</p> Signup and view all the answers

Which of the following statements about divisibility is true?

<p>If a divides b and b divides c, then a divides c. (C)</p> Signup and view all the answers

How is the concept of integers utilized in the proof?

<p>To apply properties like associativity in calculations. (A)</p> Signup and view all the answers

What does the symbol ⇨ signify in the mathematical proof?

<p>An implication. (C)</p> Signup and view all the answers

What are the positive divisors of 21?

<p>1, 3, 7, 21 (A)</p> Signup and view all the answers

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

<p>3 (D)</p> Signup and view all the answers

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

<p>a | c (B)</p> Signup and view all the answers

Which of the following statements about divisors is true?

<p>If a | b and b | a, then a = b (C)</p> Signup and view all the answers

What are the divisors of -21?

<p>1, 3, 7, 21, -1, -3, -7, -21 (A)</p> Signup and view all the answers

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

<p>There exists an integer <em>m</em> such that <em>a</em> × <em>m</em> = <em>b</em>. (A)</p> Signup and view all the answers

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

<p><em>a</em> ∤ <em>b</em> (A)</p> Signup and view all the answers

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

<p>100 is a multiple of 5. (B)</p> Signup and view all the answers

Which of the following statements is true regarding divisors?

<p>Divisors are interchangeable terms with factors. (A)</p> Signup and view all the answers

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

<p>21 is both a multiple and a divisor of itself. (B)</p> Signup and view all the answers

Flashcards

Divisibility Rule (Theorem 1.2(c))

If 'a' divides 'b' and 'c' divides 'd', then 'a' times 'c' divides 'b' times 'd'.

9 divides 5544

99 is a factor of 5544.

8 divides 1176

168 is a factor of 1176.

72 = 9 x 8

The number 72 can be expressed as the product of 9 and 8.

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Divisibility of a Product

If a number divides another number, then it must be a factor/divisor of that number.

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Divisors of 42

The positive integers that divide 42 evenly without a remainder.

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Theorem (1.2)(b)

If 'a' divides 'b' and 'b' divides 'c', then 'a' divides 'c'.

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Tau function (Ï„(n))

Counts the number of positive divisors of a positive integer 'n'.

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Positive divisors

Divisors that are greater than zero.

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Factors of 42

Numbers that can be divided into 42 without a remainder

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GCD

The greatest common divisor (GCD) of two integers is the largest positive integer that divides both numbers without leaving a remainder.

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Divisibility

One integer is divisible by another if the division produces no remainder.

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If a divides 1

If a divides 1, then a must be equal to ±1

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a divides b, b divides c

If 'a' divides 'b', and 'b' divides 'c', then 'a' divides 'c'.

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a divides b and b divides a

If a divides b and b divides a, then a = ±b

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Integers

Whole numbers, positive, negative, or zero.

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Definition (1.1)

Missing definition crucial for understanding the proof.

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Divisibility

An integer 'a' divides an integer 'b' (written as 'a | b') if there exists another integer 'm' such that 'a' multiplied by 'm' equals 'b'.

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a | b

'a' divides 'b'. There's an integer 'm' where 'a' multiplied by 'm' equals 'b'.

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Divisor

A number that divides another number completely (without a remainder).

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Factor

A number that divides another number completely (without a remainder).

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Multiple

A number that can be obtained by multiplying another number by an integer.

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Divisors

Integers that can divide another integer without remainder, also known as factors.

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Divisors of 21

The positive and negative integers that divide 21 exactly, including 1, 3, 7, 21, -1, -3, -7, -21.

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Divisors of -21

Identical to the divisors of 21, both positive and negative integers that divide -21 without remainder.

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Divisors of 100

The positive and negative integers dividing 100 completely, including ±1, ±2, ±4, ±5, ±10, ±20, ±25, ±50, ±100.

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Divisors of 1

Only positive and negative 1, ±1

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Divisors of 15

The positive and negative integers that divide 15 evenly, ±1, ±3, ±5, ±15.

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Divisors of 30

The positive and negative integers that divide 30 evenly: ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30

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a | 1

Integer a divides 1 if and only if a equals ±1.

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Transitive Divisibility

If a divides b and b divides c, then a divides c.

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Product Divisibility

If a divides b and c divides d, then the product of a and c divides the product of b and d.

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Divisor Equality

Integer a divides b, and b divides a, then a = ±b.

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Divisor Magnitude

If a divides b, the absolute value of a is less than or equal to the absolute value of b.

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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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