Vedic Patterns G Strategy | Quadratic Equations | Hard
If m and n are integers such that (m + 2n)(2m + n) = 27,
then the maximum possible value of 2m − 3n is ___
Answer & Explanation
Given: (m + 2n)(2m + n) = 27
Since m and n are integers, (m + 2n) and (2m + n) must be integer factors of 27.
Possible factor pairs (m + 2n, 2m + n) are:
(1, 27), (3, 9), (9, 3), (27, 1),
(−1, −27), (−3, −9), (−9, −3), (−27, −1)
We now solve each pair as a system: m + 2n = A; 2m + n = B
From these:
Multiply first equation by 2: 2m + 4n = 2A
Subtract from second: (2m + n) − (2m + 4n) = B − 2A
⇒ −3n = B − 2A ⇒ n = (2A − B) / 3
Then m = A − 2n
| m + 2n | 2m + n | n = (2A − B)/3 | m = A − 2n | 2m − 3n | Valid / Reject |
| 1 | 27 | (2 − 27)/3 = −25/3 | — | — | Reject |
| 3 | 9 | (6 − 9)/3 = −1 | 3 − 2(−1) = 5 | 13 | Valid |
| 9 | 3 | (18 − 3)/3 = 5 | 9 − 10 = −1 | −17 | Valid |
| 27 | 1 | (54 − 1)/3 = 53/3 | — | — | Reject |
| −1 | −27 | (−2 + 27)/3 = 25/3 | — | — | Reject |
| −3 | −9 | (−6 + 9)/3 = 1 | −3 − 2 = −5 | −13 | Valid |
| −9 | −3 | (−18 + 3)/3 = −5 | −9 + 10 = 1 | 17 | Valid |
| −27 | −1 | (−54 + 1)/3 = −53/3 | — | — | Reject |
Maximum possible valid value of 2m − 3n = 17
Vedic Patterns G Strategy | Moderate
Suppose a, b, c are three distinct natural numbers, such that 3ac = 8(a + b).
Then, the smallest possible value of 3a + 2b + c is _______
Answer & Explanation
Final Answer: 12
Given: 3ac = 8(a + b) ⇒ 8b = 3ac − 8a ⇒ b = (3ac/8) − a
Since a, b, c are natural numbers, 3ac must be divisible by 8.
Now try small values of a (since we want the minimum value of 3a + 2b + c).
Try a = 1
Then 3c must be divisible by 8 ⇒ c = 8
b = (3×1×8)/8 − 1 = 3 − 1 = 2
Value = 3a + 2b + c = 3 + 4 + 8 = 15
Try a = 2
Then 6c divisible by 8 ⇒ c = 4 is the smallest possible
b = (3×2×4)/8 − 2 = 3 − 2 = 1
Here a, b, c = 2, 1, 4 are distinct natural numbers
Value = 3a + 2b + c = 6 + 2 + 4 = 12
Try a = 3
Then 9c divisible by 8 ⇒ c = 8
b = (3×3×8)/8 − 3 = 9 − 3 = 6
Value = 9 + 12 + 8 = 29 (larger)
Hence the smallest possible value occurs at a = 2, b = 1, c = 4
