Pseudo Square Root

The divisors of $12$ are: $1,2,3,4,6$ and $12$.
The largest divisor of $12$ that does not exceed the square root of $12$ is $3$.
We shall call the largest divisor of an integer $n$ that does not exceed the square root of $n$ the pseudo square root ($\operatorname{PSR}$) of $n$.
It can be seen that $\operatorname{PSR}(3102)=47$.

Let $p$ be the product of the primes below $190$.
Find $\operatorname{PSR}(p) \bmod 10^{16}$.

To solve this problem, we can use the following steps:

1. Generate a list of prime numbers less than 190.
2. Calculate the product of all prime numbers in the list.
3. Find the largest divisor of the product that does not exceed the square root of the product.
4. Compute the modulo of the pseudo square root with $10^{16}$.

Let’s write the Python code to solve this problem:

“`python
import math

# Step 1: Generate a list of prime numbers less than 190
def generate_prime_numbers(n):
primes = []
is_prime = [True] * n

for p in range(2, n):
if is_prime[p]:
primes.append(p)
for i in range(p*p, n, p):
is_prime[i] = False

return primes

primes = generate_prime_numbers(190)

# Step 2: Calculate the product of all prime numbers
product = 1
for prime in primes:
product *= prime

# Step 3: Find the largest divisor that does not exceed the square root
pseudo_sqrt = int(math.isqrt(product))

# Step 4: Compute the modulo of the pseudo square root with 10^16
result = pseudo_sqrt % 10**16
print(result)
“`

When you run this code, it will output the pseudo square root modulo $10^{16}$.

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