Mirrored Product?

Firstly, $2460301056 = 2^{8} \times 3^{2} \times 233 \times 4583$ .

Since both $m$ and $n$ have the same number of digits, they must have $5$ digits each.

Also, one of the factors is $4583$ , and $4583 \times 233 > 99999$ .

So one of the numbers (say $m$ ) must be of the form $4583a$ where $a$ is an integer and $n$ must be of the form $233b$ where $b$ is an integer.

Since $\frac{99999}{4583} < 22$ and $\frac{10000}{4583}>2$ , $a$ can only take the value of $2^{2}$ , $2^{3}$ , $2^{4}$ , $2 \times 3$ , $2^{2} \times 3$ and $2 \times 3^{2}$ .

Testing, we find that $a=2 \times 3 = 6$ yields $m=27498$ and subsequently $n=233 \times (2^{7})(3)=89472$ . We have our desired answer $27498+89472=\boxed{116970}$ .

Since the digit is a 10 digit number and the last digit is not 0, both the numbers must be 5-digits, so I tried for integer all the way to 99999 but it only takes 1 second. Here is my Python solution: ;)

def reverse_int(number): #function to reverse an integer a = 0 b = 0 c = 0 d = 0 e = 0 while number - 10000>=0: e = e + 1 number = number - 10000 while number - 1000>=0: a = a + 1 number = number - 1000 while number - 100>=0: b = b + 1 number = number - 100 while number - 10>=0: c = c + 1 number = number - 10 while number - 1>=0: d = d + 1 number = number - 1 reverse = 10000 d + 1000 c + 100 b + 10 a + e return reverse

for i in range(99999): #testing for integer up to 99999 k = i k = reverse_int(k) if i*k == 2460301056: print(str(i) + " and " + str(k)) else: pass

So you get: 27498 and 89472 and the sum is 116970.

... Here's a hint: The two numbers are 27498 and 89472. I'm so sorry but that's all I can say; the rest is up to the solver to figure out. (Good luck finding it!)

Can't spoil the fun, can I? ;D

SCRATCH2

This is a dreadful brute force hack. Surely it's only necessary to check for solutions somewhere closer to the square root of mn?

Python:

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def re(n):   #i'm making reverse function as re(n)
    return int(str(n)[::-1])
for m in range(1,99999):
    k=re(m)
    if k*m==2460301056.0:
        print 'answer is',k+m