Large powers

$3^{12} + 2^{14} = 3^{12} +4*8^{4}$ . Using Sophie Germain identity :

$3^{12} + 4*8^{4} = (3^{6}+2^{7}+2^{4}3^{3})(3^{6}+2^{7}-2^{4}3^{3}) \Rightarrow (1289)*(425)$

1289 is prime and 425 factors into $5^{2} * 17$

$3^{12} + 2^{14} = 2189 * 5^{2} * 17 \Rightarrow 2189 + 5 + 17 = 1311 \rightarrow \boxed{311}$

The only method I know how is to actually compute that value.

$\begin{aligned} 3^{12} & = & (3^4)^3 \\ & = & 81^3 \\ & = & (80 + 1)^3 \\ & = & 1000 \cdot 8^3 + 3 \cdot 100 \cdot 8^2 + 3 \cdot 80 + 1 \\ & = & 512000 + 19200 + 240 + 1 \\ & = & 531441 \\ \end{aligned}$

And

$\begin{aligned} 2^{14} & = & (2^7)^2 \\ & = & (128)^2 \\ & = & (100 + 28)^2 \\ & = & 10000 + 2800 \cdot 2 + 28^2 \\ & = & 16384 \\ \end{aligned}$

Thus $3^{12} + 2^{14} = 547825$ , since the last two digits is $25$ and the third last digit is not $1$ or $6$ , then it's divisible by $5^2$ . Do long division, we get $\frac {547825}{25} = 21913$

The next few prime number (after $2,3,5$ ) on the List of Primes is $7,11,13,17$

Trial and error shows that only one of them divides $21913$ , which is $17$ . Do long division gives $\frac{21913}{17} = 1289$

We see that $1289$ is in the List of Primes. So the prime factorization of the expression is $5^2 \times 17 \times 1289$ . Answer is $( 5 + 17 + 1289) \bmod{1000} = \boxed{311}$

USE CALCULATOR

3^12+2^14=(3^6+2^7)^2-2^8 3^6=(3^6+2^7-2^4 3^3)(3^6+2^7+2^4 3^3)=(3^6+2^6-2 2^3 3^3+2^6)(3^6+2^6+2 2^3 3^3+2^6) =[(3^3-2^3)^2+2^6][(3^3+2^3)^2+2^6]=425 1289=5^2 17 1289 is the prime factorization. so the sum of the distinct factors=5+17+1289=1311 whose last three digits are 311.

Use SOPHIE GERMAIN'S IDENTITY and the check that 1289 is a prime.You may refer to the given link.

_ LINK _ : http://en.wikipedia.org/wiki/Sophie_Germain

3^12 + 2^14 = 5^2 x 17 x 1289. So, the distinct prime factors of 3^12 + 2^14 are 5, 17, 1289. Then, the sum is 5 + 17 + 1289 = 1311. Thus, the last three digits is 1[311] = 311. Answer : 311

Since $3^12$ is $531441$ and $2^14$ is $16384$ , the sum is $531441 + 16384 = 547825$ . Now, by using prime factorisation, $547825$ can be factorised down to $5^2 * 17 * 1289$ . Now, all we need to do is find the sum of $5$ , $17$ and $1289$ , which is $1311$ .

Knowing $2^7 = 128$ , we can calculate its square to find $2^{14} = 16384$ . Knowing $3^6 = 729$ , we can calculate its square to find $3^{12} = 531441$ . Thus, $3^{12} + 2^{14} = 547825$ .

Note that this number ends with $25$ . This means it is divisible by $25$ and thus, one of its prime factors is $5$ .

After dividing our number, we have a new ogre to analyse: $21913$ . By brute force, we can find it is not divisible by $2,3,5,7,11$ or $13$ , but it is by $17$ ; another prime factor.

After dividing again our number, we are left with $1289$ . Being its square root smaller than $36$ , we only need to analyse its divisibility by primes under $36$ . Sad news is that none of them divide $1289$ , meaning it is another prime factor.

Thus, the sum of the distinct prime factors is $1311$ , last three digits being $\boxed{311.}$ .