Two numbers have a product of 2028 and an H.C.F of 13. How many such pairs exist__________?
Show answer Hide answer
Answer: B. 2
Explanation
Expressing numbers as thirteen x and thirteen y where x and y are coprime yields a product of one sixty nine xy. Dividing two thousand twenty eight by one sixty nine gives the product of coprime variables xy equal to twelve. Identifying coprime pairs that multiply to twelve yields one and twelve as well as three and four making two pairs. Any two numbers can be expressed as their HCF multiplied by coprime variable factors. The product of two numbers equals the product of their HCF squared multiplied by their coprime factors. Coprime integers share no common positive integer factors other than the number one.
Number theory problems involving HCF and product constraints test understanding of prime factorization. Cryptographic algorithms rely on coprime properties for generating secure public and private keys. Finding valid integer pairs is essential in combinatorial mathematics and discrete problem-solving. Option A suggests exactly one pair which misses the second valid coprime combination. Option C suggests three pairs which overcounts the available coprime factors of twelve. Option D suggests four pairs which incorrectly includes non-coprime combinations like two and six.
- Subject
- Mathematics
- Category
- Mathematics
- Topic
- —
- Difficulty
- medium
Related questions
Q1. Two numbers have a product of 2160 and an HCF of 12. Find the numbers________?
Show answer Hide answer
Answer: A. (36,60)
Let the numbers be represented as twelve x and twelve y based on the given highest common factor of twelve. The product of the numbers is one forty four times x times y, which equals twenty one sixty, making x times y eq...
Q2. A town's population grew from 1,75,000 to 2,62,500 over a decade. What is the average annual percentage increase:________?
Show answer Hide answer
Answer: B. 5%
The population increase of eighty seven thousand five hundred over the base represents a fifty percent total growth. Dividing the total fifty percent growth over the ten year duration yields an average of five percent an...
Q3. An item's price is reduced by 25%. To restore its original price, by what percentage must the new price increase________?
Show answer Hide answer
Answer: A. 33 1/3 %
A twenty five percent price reduction results in a new reduced price equal to seventy five percent of the original. To restore the price back to the original level, a twenty five unit increase is needed on a seventy five...
Q4. How much water must be added to dilute 9 ml of a lotion containing 50% alcohol down to 30% alcohol_________?
Show answer Hide answer
Answer: D. 6 ml
The initial amount of pure alcohol in nine milliliters of lotion is calculated by multiplying nine by zero point five. Setting up the dilution equation with an unknown added water volume x gives four point five divided b...
Q5. Two numbers have a product of 2025 and an HCF of 15. What is their LCM____________?
Show answer Hide answer
Answer: C. 135
The fundamental property states that the product of two numbers equals the product of their HCF and LCM. Dividing the given number product of twenty zero twenty five by the HCF of fifteen gives the LCM. Executing the div...
Q6. A circular wheel has a 17.5 m radius. How many rotations does it perform over an 11 km distance?
Show answer Hide answer
Answer: C. 1000
The circumference is 2 * (22/7) * 17.5 = 110 meters. The number of rotations in 11 km (11,000 meters) is 11,000 / 110 = 100.