## The radical of $n$, $\operatorname{rad}(n)$, is the product of distinct prime factors of $n$. For example, $504 = 2^3 \times 3^2 \times 7$, so $\operatorname{rad}(504) = 2 \times 3 \times 7 = 42$.

We shall define the triplet of positive integers $(a, b, c)$ to be an abc-hit if:

$\gcd(a, b) = \gcd(a, c) = \gcd(b, c) = 1$

$a \lt b$

$a + b = c$

$\operatorname{rad}(abc) \lt c$

For example, $(5, 27, 32)$ is an abc-hit, because:

$\gcd(5, 27) = \gcd(5, 32) = \gcd(27, 32) = 1$

$5 \lt 27$

$5 + 27 = 32$

$\operatorname{rad}(4320) = 30 \lt 32$

It turns out that abc-hits are quite rare and there are only thirty-one abc-hits for $c \lt 1000$, with $\sum c = 12523$.

Find $\sum c$ for $c \lt 120000$.

### To solve this problem, we can follow these steps:

1. Create a helper function to compute the radical of a number. This function will take a number as input and return its radical. We can use prime factorization to compute the radical.

2. Implement a method to generate triplets of numbers that satisfy the given conditions. We can start by generating pairs of coprime numbers (a, b) such that a < b. Then, we can compute c = a + b and check if the radical of abc is less than c.
3. Iterate through all possible values of a < b < c < 120,000 to find the abc-hits.
4. Sum up the values of c for all the abc-hits found.
Now let's write the Python code to solve the problem:
```python
import math
def compute_radical(n):
# Helper function to compute the radical of a number
rad = 1
for i in range(2, int(math.sqrt(n)) + 1):
if n % i == 0:
rad *= i
while n % i == 0:
n //= i
if n > 1:

rad *= n

return rad

def find_abc_hits(limit):

abc_hits = []

for a in range(1, limit):

for b in range(a + 1, limit – a):

c = a + b

if c >= limit:

break

if math.gcd(a, b) == 1 and math.gcd(a, c) == 1 and math.gcd(b, c) == 1 and compute_radical(a * b * c) < c:
abc_hits.append(c)
return abc_hits
limit = 120000
abc_hits = find_abc_hits(limit)
sum_c = sum(abc_hits)
print("Sum of c for c < 120,000: ", sum_c)
```
When you run this code, it will output the sum of all `c` values for `c < 120,000` that satisfy the conditions of being an abc-hit. This may take some time to run since the computation involves a large range of numbers, but it should give you the correct answer.

##### More Answers:

Ordered RadicalsPalindromic Sums

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