Number System Questions for CAT XAT & MBA

Table of Contents

Number of Digits in Different Bases – Practice Questions

Question 1: Base 6 and Base 5 Digit Range

How many numbers are there which, when converted into base 6 and base 5, will have 5 and 6 digits respectively?

For base 6: \(6^4 \leq N < 6^5 \Rightarrow 1296 \leq N < 7776\)
For base 5: \(5^5 \leq N < 5^6 \Rightarrow 3125 \leq N < 15625\)
Intersection: \(3125 \leq N < 7776\)
Count: \(7776 - 3125 = \boxed{4651}\)

Question 2: Easier Version (Base 4 and Base 3)

How many numbers are there which, when converted into base 4 and base 3, will have 3 digits and 4 digits respectively?

Base 4: \(4^2 \leq N < 4^3 \Rightarrow 16 \leq N < 64\)
Base 3: \(3^3 \leq N < 3^4 \Rightarrow 27 \leq N < 81\)
Intersection: \(27 \leq N < 64\)
Count: \(64 - 27 = \boxed{37}\)

Question 3: Tougher Version (Base 7, Base 5, Base 3)

How many numbers are there which, when converted into base 7, base 5, and base 3, will have 4 digits, 5 digits, and 7 digits respectively?

Base 7: \(7^3 \leq N < 7^4 \Rightarrow 343 \leq N < 2401\)
Base 5: \(5^4 \leq N < 5^5 \Rightarrow 625 \leq N < 3125\)
Base 3: \(3^6 \leq N < 3^7 \Rightarrow 729 \leq N < 2187\)
Intersection: \(729 \leq N < 2187\)
Count: \(2187 - 729 = \boxed{1458}\)

Question 4: Unique Number (Only One Common)

Find the number which, when converted into:

base 9 has 3 digits,
base 7 has 4 digits,
base 5 has 5 digits.

Assume only one such number exists. What is that number?

✅ Correct Answer: Option B. 625

Base 9: \(625 < 729 \Rightarrow\) 3-digit
Base 7: \(625 > 343 \Rightarrow\) 4-digit
Base 5: \(625 = 5^4 \Rightarrow\) 5-digit
It satisfies all three conditions, and only this number lies in the intersection range.
Hence, the only valid number is: \(\boxed{625}\)