The numbers 525 and 3000 are both divisible only by 3, 5, 15, 25 and 75. What is HCF (525, 3000)? Justify your answer.
HCF = 75.
Step 1: Prime factorisation of 525.
Divide step by step:
\(525 \div 3 = 175\) ⇒ so one factor is \(3\).
\(175 \div 5 = 35\) ⇒ factor \(5\).
\(35 \div 5 = 7\) ⇒ another factor \(5\).
So, \(525 = 3 \times 5^2 \times 7\).
Step 2: Prime factorisation of 3000.
Divide step by step:
\(3000 \div 2 = 1500\)
\(1500 \div 2 = 750\)
\(750 \div 2 = 375\)
\(375 \div 3 = 125\)
\(125 \div 5 = 25\)
\(25 \div 5 = 5\)
\(5 \div 5 = 1\)
So, \(3000 = 2^3 \times 3 \times 5^3\).
Step 3: Identify common prime factors.
From 525: \(3, 5^2, 7\).
From 3000: \(2^3, 3, 5^3\).
Common factors are:
\(3\) and \(5^2\).
Step 4: Multiply common factors to get HCF.
\(HCF = 3 \times 5^2 = 3 \times 25 = 75\).
Conclusion. The highest common factor (HCF) of 525 and 3000 is 75.