Question:
How many natural numbers, when converted to base 3, are four-digit numbers, but when converted to base 4, are three-digit numbers ?
How many natural numbers, when converted to base 3, are four-digit numbers, but when converted to base 4, are three-digit numbers ?
Updated On: Aug 21, 2024
- 27
- 37
- 41
- 47
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The Correct Option is B
Solution and Explanation
Let ‘x’ be the total number of natural numbers.
Since x has four-digit numbers in base 3, 33 ≤ x < 34 or 27 ≤ x < 81.
Also, x has three-digit numbers in base 4, 42 ≤ x < 43 or 16 ≤ x < 64.
Thus, the common part of the ranges of x will be 27 ≤ x < 64.
So, the desired answer is (64 – 27) = 37 numbers.
So, the correct option is (B) : 37.
Since x has four-digit numbers in base 3, 33 ≤ x < 34 or 27 ≤ x < 81.
Also, x has three-digit numbers in base 4, 42 ≤ x < 43 or 16 ≤ x < 64.
Thus, the common part of the ranges of x will be 27 ≤ x < 64.
So, the desired answer is (64 – 27) = 37 numbers.
So, the correct option is (B) : 37.
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