The sides of ΔABC are a, b, and c. If the lengths of a = 7 and b = 10, what are the possible lengths of c?

The possible lengths of c are between 3 to 17.

Step-by-step explanation:

According to the scenario, calculation of the given data are as follows,

a = 7

b = 10

As we know that, in a triangle length of third side is always greater than difference between the given two side, and always lesser than sum of both side.

So, Difference between both side = 10 - 7 = 3

Sum of both side = 10 + 7 = 17

Hence, the possible lengths of c are between 3 to 17.

16

Step-by-step explanation:

We must find integers x, y with the most amount of prime divisors, not necessarily distinct, such that x + 3y < 1,000.

Obviously, this is achieved when the divisor is the least prime 2. So, we must find integers n, m such that

since , then n must be 9. For n=9 we find the greatest integer m such that  

and we find m=7

 and   ,  are the numbers we are looking for and the sum of their length is 9+7 = 16.

So, 16 is the maximum possible sum of the length of x and the length of y.