The total number of -digit numbers whose greatest common divisor with is , is:
NVAEasyJEE 2023Fundamental Principle of Counting
JEE Mathematics 2023 Question with Solution
Answer
Correct answer:3000
Step-by-step solution
Standard Method
Given: We need the total number of -digit numbers whose greatest common divisor with is .
Find: The required count of such numbers.
A number must be divisible by but not by to satisfy the condition.
Total -digit numbers divisible by :
Total -digit numbers divisible by :
Therefore, numbers divisible by but not by :
Therefore, the required number is .
Common mistakes
Counting all numbers divisible by and stopping there is incorrect, because numbers also divisible by will have gcd with greater than . Exclude numbers divisible by instead.
Using the condition gcd as only 'not divisible by ' is wrong. The number must share exactly one factor of with and no factor of , so test divisibility by and carefully.
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