Find the greatest common divisor of 504 and 385.

Question

BlackTom AI · Accepted Answer

Applying the Euclidean algorithm gives the fastest route to the greatest common divisor (GCD).

1. Relevant knowledge points and concepts
- Greatest common divisor (GCD) of two integers is the largest integer dividing both without remainder.
- Euclidean algorithm: gcd(a, b) = gcd(b, a mod b); repeat until remainder 0. The last nonzero remainder is the gcd.
- Prime factorization: GCD can also be found by multiplying the common prime factors with the smallest exponents.

2. Complete reasoning process (step by step using Euclidean algorithm)
- Start with the pair (504, 385).
- Compute 504 mod 385: 504 − 385 = 119. So gcd(504, 385) = gcd(385, 119).
- Compute 385 mod 119: 119 × 3 = 357, and 385 − 357 = 28. So gcd(385, 119) = gcd(119, 28).
- Compute 119 mod 28: 28 × 4 = 112, and 119 − 112 = 7. So gcd(119, 28) = gcd(28, 7).
- Compute 28 mod 7: 7 × 4 = 28, remainder 0. When remainder is 0, the last nonzero remainder (7) is the gcd.

3. Intermediate calculation steps (explicit)
- 504 = 385·1 + 119
- 385 = 119·3 + 28
- 119 = 28·4 + 7
- 28 = 7·4 + 0
Thus gcd = 7.

4. Why the correct answer is correct
- The Euclidean algorithm leads unambiguously to 7 as the last nonzero remainder, so 7 divides both numbers and no larger integer divides both.
- Prime-factor check: 504 = 2^3 · 3^2 · 7 and 385 = 5 · 7 · 11. The only common prime factor is 7, so GCD = 7.

5. Why other possible answers would be incorrect
- Any number >7 cannot divide both because the prime factorizations share only one common prime 7 (so a larger common divisor would require additional common primes).
- Numbers like 14, 21, 35 fail because they include extra prime factors (2,3,5,11) not common to both 504 and 385.

6. Conclusion confirming the final answer
- Therefore, the greatest common divisor of 504 and 385 is 7, which matches the provided answer.

Find the greatest common divisor of 504 and 385.Short answer

Similar Questions

For the compound below:a. Give the IUPAC name (no stereochemical designation required. [1 mark]b. Draw the two possible chair conformers of the compound and use equilibrium arrows to show which form will predominate. Explain your selection. [4 marks][Fill in the blank]

If a sample in a UV spectrometer transmits 20% of the incident radiation, the absorbance is:[Fill in the blank]

What would be the expected product from the following reaction?[Fill in the blank]

Alcohols can:[Fill in the blank]

Which of the following is both a weak base and a strong nucleophile?[Fill in the blank]

By which of these mechanism(s) would the following reaction proceed most favourably?[Fill in the blank]

Which of the following names describes the relationship between these two structures?[Fill in the blank]

How many stereoisomers are possible for the structure below?[Fill in the blank]

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler