Have fun with GCD & LCM!

If the two numbers have 14 as GCD, then they can be expressed in the form 14x and 14y where x and y are coprime. Then, their LCM will be 14xy. ---> 14xy=6468 ---> xy=462 -------> Eq 1 Now, m+n=14(x+y) From Eq 1, solving for x and y using the factors of 462 the minimum value of x+y is 43 Thus, the minimum value of m+n will be 14(x+y)=14(43)=602

We've GCD and LCM are 14 [2 . 7] and 6468 [2^2 . 3 . 7^2 . 11]. because GCD is [ 2 . 7], then m and n that meet are {2.7.2.7 and 2.7.3.11},{3.7.2.7 and 2.2.7.11},{3.7.7.11 and 2. 2. 2.7}, {etc}. But smallest m and n is 3.7.2.7 and 2.2.7.11 ---> m = 3.7.2.7 = 294. n = 2.2.7.11 = 308 . m + n = 294 + 308 = 602. Answer : 602

We know that $lcm (m, n)$ $= \frac {m\cdot n}{\gcd (m, n)}$ . Substituting what we have, we get $14\cdot6468 = m\cdot n$ . Checking carefully the pairs of possible $m$ and $n$ , we find that the smallest $m+n$ is $294$ and $308$ , where $294 + 308 = \boxed {602}$ .

All those methods are awesome but what I did is the produxt of the numbers must be 14×6468=90552

So I did √90552 and multiplied it by 2 which got me 601.83 and I rounded it to 602 and that's your answer :-D

We know that the product of the greatest common divisor of two integers and the least common multiple of the same two integers is simply the product of the integers. Therefore, $m \cdot n = 6468 \cdot 14 = 90552$ . To minimize $m+n$ , we must bring $m$ and $n$ as close together as possible without violating the condition $m \cdot n = 90552$ . The square root of $90552$ is about $300$ , so we treat $300$ as our comparison point and look for the pair of divisors of $90522$ that are closest to $300$ . The closest pair of divisors is $(294,308)$ , so $m+n = \boxed{602}$ .

1, 2, 3, 4, 6, 7, 8, 11, 12, 14, 21, 22, 24, 28, 33, 42, 44, 49, 56, 66, 77, 84, 88, 98, 132, 147, 154,

168, 196, 231, 264, 294, 308, 343, 392, 462, 539, 588, 616, 686, 924, 1029, 1078, 1176, 1372, 1617,

1848, 2058, 2156, 2744, 3234, 3773, 4116, 4312, 6468, 7546, 8232, 11319, 12936, 15092, 22638

, 30184, 45276, 90552

are the factors in which 308*294=90552 is smallest.

so answer is 605

We know, LCM(m,n) * HCF(m,n) = m * n Therefore m * n = 14 * 6468 = 90552. 2 numbers whose product is 90552 and sum will be least, will be the ones closest to the approx square root of 90552 ie. 300. By factorisation, we see that 90552=294 * 308 = m * n Thus m+n = 294+308= ""602""

The product of the two numbers is the product of their GCD and LCM. Given $mn$ we can minimize $m+n$ by minimizing $m-n$ (assuming without loss of generality $m>n$ )as $(m+n)^2-(m-n)^2=4mn$ . Now $14\times 6468=7^3 \times 2^3\times 3\times 11$ and we can take $m=2^2\times 7\times 11=308$ and $n=2\times 3\times 7^2=294$ to minimize $m-n$

Let m = 14a and n = 14b(a does not divide b as 14 is g.c.d). Product of two numbers = g.c.d * l.c.m mn = 14 * 6468 196ab = 14 * 6468 ab = 462 = 2 3 7*11. Minimum value of a+b = 43(At a = 22 and b= 21 and vice versa) Hence, m+n=14 * 43 = 602