The number of 525 and 3000 are divisible by 3,5,15,25 and 75 what is the HCF of 525 and 3000?

Answer

To find the HCF (Highest Common Factor) of the numbers 525 and 3000, we can follow these steps:

### Step 1: Prime Factorization of 525
To find the HCF, we first need to find the prime factorization of 525.

1. Divide 525 by 5 (the smallest prime factor):
\[ 525 \div 5 = 105 \]
2. Next, factor 105:
\[ 105 \div 3 = 35 \]
3. Finally, factor 35:
\[ 35 \div 5 = 7 \]

So, the prime factorization of 525 is:
\[ 525 = 5^2 \times 3^1 \times 7^1 \]

### Step 2: Prime Factorization of 3000
Now, let's find the prime factorization of 3000.

1. Divide 3000 by 10:
\[ 3000 \div 10 = 300 \]
2. Factor 300:
\[ 300 \div 10 = 30 \]
3. Factor 30:
\[ 30 \div 10 = 3 \]

Now, we can express 3000 as:
\[ 3000 = 3^1 \times 10^3 = 3^1 \times (2 \times 5)^3 = 3^1 \times 2^3 \times 5^3 \]

So, the prime factorization of 3000 is:
\[ 3000 = 2^3 \times 3^1 \times 5^3 \]

### Step 3: Identify Common Factors
Now we will identify the common prime factors from both factorizations.

- For \(3\): The minimum power is \(3^1\).
- For \(5\): The minimum power is \(5^2\).
- \(2\) is not a common factor since it only appears in 3000.

### Step 4: Calculate HCF
Now, we can calculate the HCF by multiplying the common factors with their lowest powers:
\[ \text{HCF} = 3^1 \times 5^2 = 3 \times 25 = 75 \]

### Conclusion
Thus, the HCF of 525 and 3000 is:
\[ \text{HCF} = 75 \]

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