Highest common factor Write each number using prime factors. Find the HCF of each number pair. 525 = ? × ? × ? × ? 385 = ? × ?
The Correct Answer and Explanation is:
Prime Factorization and Highest Common Factor (HCF)
Let’s begin by writing each number as a product of its prime factors.
Step 1: Prime factorization
525
We start dividing 525 by the smallest prime numbers:
- 525 ÷ 3 = not a whole number
- 525 ÷ 5 = 105
- 105 ÷ 5 = 21
- 21 ÷ 3 = 7
- 7 is a prime number
So the prime factorization of 525 is:
525 = 5 × 5 × 3 × 7
385
We divide 385 by prime numbers:
- 385 ÷ 5 = 77
- 77 ÷ 7 = 11
- 11 is a prime number
So the prime factorization of 385 is:
385 = 5 × 7 × 11
Step 2: Find the HCF (Highest Common Factor)
Now, compare the prime factors of both numbers:
- 525 = 5 × 5 × 3 × 7
- 385 = 5 × 7 × 11
The common prime factors are 5 and 7.
Now, multiply the common prime factors:
HCF = 5 × 7 = 35
Explanation
The Highest Common Factor (HCF) is the largest number that divides two or more numbers exactly without leaving a remainder. One of the most reliable ways to find the HCF is by using prime factorization. This involves breaking each number down into the prime numbers that multiply together to make the original number.
In this case, we began by finding the prime factors of 525. We did this by starting with the smallest prime numbers. Since 525 ends in a 5, we could divide by 5, giving us 105. We repeated the process with 5 again, resulting in 21. Then, 21 was divisible by 3, and finally, we were left with 7, which is also a prime number. So, 525 is made of the prime factors 5, 5, 3, and 7.
Next, we broke 385 into prime factors. It is also divisible by 5, which gives us 77. Then 77 is divisible by 7, giving us 11. All of these are prime numbers, so the prime factorization of 385 is 5, 7, and 11.
To find the HCF, we look at the prime factors that are common to both numbers. Both 525 and 385 include the numbers 5 and 7 in their factorizations. Multiplying these gives us the HCF. Therefore, the greatest number that divides both 525 and 385 exactly is 35.
Understanding prime factorization helps in various mathematical topics like simplifying fractions, finding least common multiples, and solving ratio problems. It is a foundational skill in number theory and arithmetic.
