Two numbers sum to 528 and have an H.C.F of 33. How many pairs of numbers satisfy these conditions__________?
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Answer: A. 4
Explanation
Expressing the two numbers as multiples of their highest common factor introduces two coprime variables. Dividing the total sum by the common factor gives the sum of the coprime variables equal to sixteen. Identifying all pairs of coprime integers that sum to sixteen yields exactly four valid combinations. Any two numbers sharing a highest common factor can be written as that factor multiplied by coprime integers. Coprime integers are numbers that share no positive integer divisors other than one. Determining possible number pairs requires listing all combinations of coprime integers that satisfy the target sum.
Number theory applications use highest common factor properties to simplify fraction reductions and divisibility tests. Cryptographic algorithms rely on coprime properties to generate secure public and private key pairs. This mathematical technique is essential for solving advanced integer programming and divisibility puzzles. Option B overcounts the number of valid coprime pairs by including combinations that share common factors. Option C assumes a larger number of valid combinations by ignoring the strict coprime requirement. Option D represents an incorrect count that fails to filter out numbers with shared common divisors.
- Subject
- Mathematics
- Category
- Mathematics
- Topic
- —
- Difficulty
- medium
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