Question

The difference between the greatest number of 4 digits and the least number of 5 digits that have 144 as their HCF is

• A. \(196\)
• B. \(168\)
• C. \(144\)
• D. \(136\)

Hint:

To find nearest multiples, divide by the given HCF and adjust to the closest lower or higher integer.

Answer 1

The Correct Option is C

Concept: If a number has HCF \(144\), then it must be a multiple of \(144\). So we need:
• Greatest 4-digit multiple of \(144\)
• Least 5-digit multiple of \(144\)

Step 1: Find the greatest 4-digit multiple of \(144\).
Largest 4-digit number: \[ 9999 \] Divide: \[ 9999 \div 144 = 69 \text{ remainder} \] Nearest multiple: \[ 144 \times 69 = 9936 \] So greatest 4-digit number: \[ 9936 \]

Step 2: Find the least 5-digit multiple of \(144\).
Smallest 5-digit number: \[ 10000 \] Divide: \[ 10000 \div 144 = 69 \text{ remainder} \] Next multiple: \[ 144 \times 70 = 10080 \] So least 5-digit number: \[ 10080 \]

Step 3: Find the difference.
\[ 10080-9936 \] \[ =144 \] Thus, the required answer is: \[ \boxed{144} \]