Class 9 Ganita Manjari
Chapter 9: Propositions and their Converses
Part 8
[cite: 6]
Exercise 9.1: Prime Factors and Co-primes
In this part, we continue examining the questions from Exercise 9.1. Number theory relies heavily on how numbers interact with each other, especially through multiplication and division. [cite: 6]
When testing if a number is divisible by a large target (like 24 or 60), we often break the target down into smaller pieces. But as we will see, you cannot just break a number into any two pieces. If those pieces share common factors, the logical converse completely falls apart! Let's uncover why.
In this part, we continue examining the questions from Exercise 9.1. Number theory relies heavily on how numbers interact with each other, especially through multiplication and division. [cite: 6]
When testing if a number is divisible by a large target (like 24 or 60), we often break the target down into smaller pieces. But as we will see, you cannot just break a number into any two pieces. If those pieces share common factors, the logical converse completely falls apart! Let's uncover why.
1 Divisibility Converses (Q8 & Q9)
Let n be a positive integer. Evaluate the following two propositions from the textbook: [cite: 6]
Question 8: "If n is divisible by 24, then it is divisible by both 4 and 6." [cite: 6]
Question 9: "If n is divisible by 60, then it is divisible by both 5 and 12." [cite: 6]
Frame the converse for both statements. Determine if the converses are true or false. Justify your answers and provide a counterexample if a statement is false.
Question 8: "If n is divisible by 24, then it is divisible by both 4 and 6." [cite: 6]
Question 9: "If n is divisible by 60, then it is divisible by both 5 and 12." [cite: 6]
Frame the converse for both statements. Determine if the converses are true or false. Justify your answers and provide a counterexample if a statement is false.
Concept:
1
A number is guaranteed to be divisible by the product of two factors ONLY IF those two factors are "co-prime" (meaning they share absolutely zero common factors other than 1). [cite: 6]
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Hint:
1
For Q8 Converse: "If a number is divisible by 4 and 6, then it is divisible by 24." Are 4 and 6 co-prime? No, they both divide by 2! Find a number that breaks this rule.
2
For Q9 Converse: "If a number is divisible by 5 and 12, then it is divisible by 60." Are 5 and 12 co-prime? Yes! [cite: 6]
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Mistakes to Avoid:
!
Assuming all products work backwards: Because 4 x 6 = 24, many people assume passing the rule of 4 and 6 means passing 24. Always check for common factors first to avoid this trap!
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Solution:
Part 1: Evaluating Question 8
- Converse: "If n is divisible by both 4 and 6, then it is divisible by 24." [cite: 6]
- Truth: False.
- Counterexample: The number 12 is perfectly divisible by 4 and by 6. However, 12 is NOT divisible by 24. Because 4 and 6 share a common factor (2), their true combined divisibility rule is their Lowest Common Multiple (12), not their product (24).
Part 2: Evaluating Question 9
- Converse: "If n is divisible by both 5 and 12, then it is divisible by 60." [cite: 6]
- Truth: True.
- Justification: Because 5 and 12 share no common factors (they are co-prime), any number that is a multiple of both MUST be a multiple of their product (\(5 \times 12 = 60\)). There are no counterexamples possible. [cite: 6]
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2 Prime Factors (Q10 & Q11)
Evaluate the following two propositions from the textbook: [cite: 6]
Question 10: "If n is the square of a prime number, then it has exactly 3 factors." [cite: 6]
Question 11: "If n is a product of two unequal prime numbers, then it has exactly 4 divisors." [cite: 6]
Frame the converse for Question 11. Is the converse true or false? Provide a clear counterexample if it is false.
Question 10: "If n is the square of a prime number, then it has exactly 3 factors." [cite: 6]
Question 11: "If n is a product of two unequal prime numbers, then it has exactly 4 divisors." [cite: 6]
Frame the converse for Question 11. Is the converse true or false? Provide a clear counterexample if it is false.
Concept:
1
A product of two unequal primes (like p and q) creates a number with exactly 4 factors: 1, p, q, and pq. [cite: 6]
2
But are these the ONLY numbers that have exactly 4 factors? We must test other structures, such as a prime number raised to the third power (cubed). [cite: 6]
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Hint:
1
Converse: "If n has exactly 4 divisors, then it is a product of two unequal prime numbers." [cite: 6]
2
Look at the number 8. What are its factors? 1, 2, 4, 8. That's exactly 4 factors!
3
Is 8 the product of two unequal primes? No, it's 2 cubed. [cite: 6]
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Mistakes to Avoid:
!
Assuming Q10 and Q11 are identical in logic: The converse for Q10 is actually true (only prime squares have exactly 3 factors). However, the converse for Q11 is false because cubes of primes also have exactly 4 factors. You must test each one individually!
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Solution:
Evaluating Question 11
- Original Proposition: "If n is a product of two unequal prime numbers, then it has exactly 4 divisors." [cite: 6] (This is True. Example: \(3 \times 5 = 15\). Factors are 1, 3, 5, 15).
- The Converse: "If n has exactly 4 divisors, then it is a product of two unequal prime numbers." [cite: 6]
- Truth: False.
- Counterexample: Consider the number 8. The divisors of 8 are 1, 2, 4, and 8 (exactly 4 divisors). However, 8 is not the product of two unequal primes; it is the cube of a single prime number (\(2^3\)). Because this alternate structure exists, the converse is broken. [cite: 6]
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Objective 1: Logic of Question 12
Consider Question 12: "If n and n + 3 have no factors in common, then n is not a multiple of 3." [cite: 6]
Is the converse of this statement universally true or false?
Is the converse of this statement universally true or false?
Concept:
1
If two numbers share a factor, that same factor MUST also divide the difference between the two numbers. [cite: 6]
2
The difference between n and n+3 is exactly 3. Therefore, the only possible common factor they could ever share is 3 itself. [cite: 6]
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Hint:
1
The converse is: "If n is not a multiple of 3, then n and n+3 have no factors in common." [cite: 6]
2
Since we proved 3 is the ONLY possible common factor, and the first part says "n is NOT a multiple of 3", is it possible for them to share anything?
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Mistakes to Avoid:
!
Testing random numbers blindly: You might test n=5 and get 5 and 8, then say "Oh it's true for 5". You need to know WHY it's true for all non-multiples of 3, which requires looking at their difference (3).
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Solution:
1. The Converse: "If n is not a multiple of 3, then n and n+3 have no factors in common." [cite: 6]
2. The Proof: The difference between the two numbers is 3. A fundamental rule of divisibility states that any common factor of two numbers must also divide their difference.
Because the difference is 3 (a prime number), the only possible common factor greater than 1 that they could ever share is 3 itself.
If we start with the condition that n is NOT a multiple of 3, it cannot have 3 as a factor. Therefore, it is mathematically impossible for the two numbers to share any factors. The converse is universally True. [cite: 6]
Correct Option: (A)
Objective 2: Factors of a Prime Square
If a mystery number has exactly 3 factors, what type of number must it be?
Concept:
1
Because 3 is an odd number, we know the mystery number must be a perfect square. [cite: 6]
2
But it only has exactly 3 factors. Those factors will be 1, its square root, and itself. For this to be true, the square root cannot have any factors of its own, meaning the square root must be prime! [cite: 6]
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Hint:
1
Look at the Notebook SVG graphic above. "Factors of 25 are: 1, 5, 25." [cite: 6]
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Mistakes to Avoid:
!
Assuming all squares have 3 factors: Do not select Option A. 36 is a perfect square, but its factors are 1, 2, 3, 4, 6, 9, 12, 18, 36 (9 factors). Only squares of primes have exactly 3!
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Solution:
If a number has an odd number of factors, it is a perfect square. [cite: 6]
The factors of a perfect square are always 1, the number itself, and all the factors that build up to it.
If there are exactly 3 factors, they can only be: 1, the square root, and the number itself. [cite: 6]
For the square root to have no other factors to contribute to the list, the square root must be a prime number. Therefore, the mystery number is the square of a prime number. [cite: 6]
Correct Option: (B)
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Extra Practice 1
Let's combine concepts from Question 10 and 11! A mystery number n has exactly 5 factors.
Based on what we know about prime squares having exactly 3 factors, and prime cubes having exactly 4 factors, what specific mathematical structure must this mystery number have to produce exactly 5 factors?
Based on what we know about prime squares having exactly 3 factors, and prime cubes having exactly 4 factors, what specific mathematical structure must this mystery number have to produce exactly 5 factors?
Concept:
1
Because 5 is an odd number, the mystery number is mathematically guaranteed to be a perfect square. [cite: 6]
2
We know a prime to the power of 1 has 2 factors. Power of 2 has 3 factors. Power of 3 has 4 factors. The pattern continues!
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Hint:
1
A prime number p has factors: 1, p.
2
p² has factors: 1, p, p².
3
p³ has factors: 1, p, p², p³.
4
What exponent will give us exactly 5 factors?
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Mistakes to Avoid:
!
Assuming it's a product of primes: If it was the product of two primes (pq), it would have 4 factors. If it was the square of a product of primes (like 36), it would have 9 factors. It must be a single prime raised to a specific power! [cite: 6]
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Solution:
1. Because the total number of factors is odd (5), the number must be a perfect square.
2. By following the pattern of exponents for a prime number p: [cite: 6]
- \(p^1\) has 2 factors.
- \(p^2\) has 3 factors.
- \(p^3\) has 4 factors.
- \(p^4\) has exactly 5 factors! (1, p, p², p³, p⁴).
Conclusion: To have exactly 5 factors, the number must be a prime number raised to the 4th power. (For example, \(2^4 = 16\), which has factors 1, 2, 4, 8, 16).
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