In order for a number to be a multiple of 10, it must end in a zero. We need to find out if 3^n + 7^n will end in a 0 in order to answer this question.
To do this we can construct a table to see what the 'unit's digit is when we put either 3^n or 7^n.
n | 3^n unit | 7 ^ n unit | Sum end in 0?
--|----------|-------------|-------------
1 | 3 | 7 | Y
2 | 9 | 9 | N
3 | 7 | 3 | Y
4 | 1 | 1 | N
5 | 3 | 7 | Y
6 | 9 | 9 | N
7 | 7 | 3 | Y
It is evident that only the odd values of N give a multiple of 10, however we can see a pattern; the unit of the number will cycle after you add 4. For example, 3^1 ends in the same digit as 3^5 and 3^9.
As a result the final digit of the sum (3^n + 7^n) will always be 1 of 4 possibilities. It is either '3 + 7', '9 + 9', '7 + 3' or '1 + 1', of which two are multiples of 10 and happen to be when n is either 1 or 3. Therefore we can see that adding any multiple of 4 to 1 or 3 will always yield an odd number - hence odd number values of n will always give multiples of 10.
We can extend this by saying that when n is in the form '4k + 2', the final digit will be an 8 (9+9=18) and when n is in the form '4k' (i.e. is a multiple of 4), it will end in 2 - Either way it will be a multiple of 2. Therefore the sum is always even.
Note: k is any integer greater or equal to 0.