Question

If HCF \((72, q)=12\) then how many values can q take? (Assume \(q\) be a product of a power of 2 and a power of 3 only) (1) 1 (2) 2 (3) 3 (4) 4

Short answer

Answer: There are 4 possible values for \(q\).

Step-by-step solution

Prime factorization of 72

For finding the prime factorization of 72, we'll divide 72 with the smallest prime numbers to get the prime factors. \(72 = 2^3 * 3^2\) The prime factorization of 72 is \(2^3 * 3^2\).

Find the HCF

The HCF of two numbers is the product of the lowest powers of common prime factors. Since the HCF\((72, q) = 12\), we need to find the prime factors of 12 as well. \(12 = 2^2 * 3^1\) Now we can see that the HCF of 72 and \(q\) has the common prime factors of both, with the lowest power of each factor. Therefore, to maintain the HCF\((72, q) = 12\), \(q\) must have the factors \(2^2 * 3^1\), which is already part of the HCF.

Calculate possible values of \(q\)

Given that \(q\) can only be a product of a power of 2 and a power of 3, let's look at the remaining powers of the prime factors in 72 after subtracting the HCF's prime factors. Remaining powers of 2 in 72 after accounting for the HCF: \(2^1\) Remaining powers of 3 in 72 after accounting for the HCF: \(3^1\) Now the only possible combinations of the remaining powers of 2 and 3 that maintain the HCF are: 1. \(2^0 * 3^0 = 1\) 2. \(2^1 * 3^0 = 2\) 3. \(2^0 * 3^1 = 3\) 4. \(2^1 * 3^1 = 6\) So there are 4 possible values for \(q\): 1, 2, 3, and 6. Therefore, the correct answer is (4) 4.

Key concepts

highest common factor (HCF)

The highest common factor (HCF), also known as the greatest common divisor (GCD), of two numbers is the largest number that divides both of them without leaving a remainder. It is a crucial concept in number theory, often used in simplifying fractions and solving problems that involve divisibility.
To find the HCF, you can use the method of prime factorization, which involves expressing each number as a product of prime numbers. Then, identify the common prime factors and select the lowest power shared by each factor as part of the HCF.

• The process helps in solving problems that require simplifying ratios or determining shared multiples, ensuring the relation between numbers remains consistent.
• Using HCF, you can understand the structure of numbers better, as it breaks them down into their basic prime components, providing a clear view of any divisibility relationships.

Understanding the HCF deepens one’s grasp of algebraic and numeric relationships, serving as a foundation for more advanced mathematical concepts.

prime factorization

Prime factorization is the process of breaking down a number into the set of prime numbers that multiply together to give the original number. Each positive integer has a unique prime factorization, paving the way to understand its fundamental building blocks.
To perform prime factorization, repeatedly divide the number by the smallest prime number until all the factors left are prime. For instance, the prime factorization of 72 is obtained by dividing by 2 and 3, resulting in 72 = 2^3 * 3^2.

• This step is essential, particularly in finding the HCF, as it reveals the specific prime numbers that can be common among different integers.
• Prime factorization not only facilitates the comprehension of how numbers relate to each other but also simplifies the calculation of the HCF by making the commonality of prime factors explicit.

It’s a core tool in mathematics for breaking down numbers into their fundamental elements.

powers of prime numbers

The power of a prime number in prime factorization refers to how many times that prime number is multiplied by itself in the factorization. It gives a clear indication of the frequency or dominance of a certain prime factor within a number.
In the case of 72, the power of 2 is 3, and the power of 3 is 2. These powers are critical when comparing numbers for common factors. For determining the HCF, it is crucial to look at the lowest power of each common prime factor.
Understanding powers within numbers aids in determining their divisibility and compatibility with other numbers.

• Powers offer a neat way to express larger calculations, as they replace repetitive multiplication with a concise numeric shorthand.
• They allow the computation of factors and divisors more easily, especially when dealing with larger numbers or multiple calculations.

Mastering the concept of powers in numbers lays the groundwork for polynomial operations and exponentiation in higher mathematics.

mathematics problem-solving step-by-step

Solving mathematical problems step-by-step is an organized approach that ensures thorough understanding and consistent results. Breaking problems into logical steps makes complex problems more manageable and clear.
Here's how a step-by-step method works to solve for a mathematical problem, such as finding possible values for q given HCF constraints:
First, identify what you are being asked to find. Here, it involves determining how many values q can take. Then, use known information to form equations or logical conclusions, like prime factorization of the given numbers.
Apply mathematical concepts, like using the HCF to find shared factors and powers, extracting all possible combinations of remaining factors.

• Following this methodical process minimizes errors, as it structures how you approach each element of a question logically.
• This approach can be applied to numerous problem-types, from factoring equations to calculating geometrical angles.

Perfecting step-by-step problem-solving techniques enhances mathematical literacy and empowers the learner to tackle increasingly advanced topics.