Permutations & Combinations
Counting numbers with given GCD
nta_pyq_2023_jan
Grade 11
Question:
The total number of 4-digit numbers whose greatest common divisor with 54 is 2, is _____.
Step-by-Step Solution
Key Concept: GCD(N,54)=2 means N is divisible by 2 but not by 3 (since 54=2$\times$3³). Count 4-digit even numbers not divisible by 3.
N = (4-digit numbers divisible by 2) − (4-digit numbers divisible by 6) = 9000/2 − 9000/6 = 4500 − 1500 = 3000.
Correct Answer: 3000