Number System Question 3 - CAT 2023
For any natural numbers m, n and k, such that k divides both m+2n and 3m+4n, k must be a common divisor of:
Step-by-Step Explanation
- Apply the linear combination property of divisibility:
If a number k divides two distinct algebraic expressions, it must also divide any linear combination of those expressions.
Given that k divides (m + 2n) and (3m + 4n):
k divides 3(m + 2n) = 3m + 6n
Take the difference between these two derived expressions:
k divides (3m + 6n) - (3m + 4n) = 2n
- Isolate the second component variable:
Similarly, since k divides (m + 2n), it must also divide its double:
k divides 2(m + 2n) = 2m + 4n
Take the difference between the second original expression and this new expression:
k divides (3m + 4n) - (2m + 4n) = m
Since k must divide both m and 2n, it is a common divisor of m and 2n.
Correct Option: C
💡 FundaMakers Tip:
Use the standard algebraic elimination method on variables directly within divisibility constraints. Multiplying one expression to match and eliminate a variable coefficient via subtraction reveals the remaining underlying factor immediately.