CAT 2023 Slot 2 QA Question 2

Multiple choice (+3 / −1) · Number Systems · HCF/LCM · Try it, then check the answer and solution below.

For any natural numbers

m, n , and

k , such that

k

divides both

m+2 n

and

3 m + 4 n, k

must be a common divisor of

⌨️ Shortcuts:[1-4 / A-D] Select[Enter] Check[B] Star[S] Solution

Official Convenor Answer Key: D

Answer and solution

Answer: D) $m$ and $2n$

📌 Core Concept

The key concept here is that if a number

k

divides two expressions, it must also divide any linear combination of those expressions. This property helps in determining the common divisors.

🔢 Step-by-Step Solution

Given:

k

divides

m + 2n

⇒

m + 2n = k \cdot a

for some integer

a

k

divides

3m + 4n

⇒

3m + 4n = k \cdot b

for some integer

b

Form Linear Combinations:

Subtract 3 times the first equation from the second:

(3m + 4n) - 3(m + 2n) = 3m + 4n - 3m - 6n = -2n

Thus,

k

divides - 2n

⇒

divides

2n$.

Subtract 2 times the first equation from the second:

(3m + 4n) - 2(m + 2n) = 3m + 4n - 2m - 4n = m

Thus,

k

divides

m

Conclusion:

k

divides both

m

and

2n

Therefore,

k

must be a common divisor of

m

and

2n

⚡ 30-Second Shortcut

k

divides both

m + 2n

and

3m + 4n , subtracting appropriate multiples of these expressions shows

divides

and

. Hence,

is a common divisor of

and

2n$.

🎯 Final Answer

Correct Answer: D

Keep going