Smallest Prime?

In order for this to happen, $4p > 2018$ , but $3p >= 2018$ . In order to minimize $p$ , we must make $4p$ as greater than, but as close to, $2018$ as possible.

This means that $p>\frac{2018}{4}$ or $p>504.5$ . $509$ is the smallest prime number greater than $504.5$ , and thus $\boxed{p=509}$ .

Just to check, the multiples of $509$ are: $509, 1018, 1527, \textcolor{#D61F06}{2036}$ ( $2036>2018$ )

Normally PHP is my go to language, but 2018! is too big for that language to handle, so I had to improvise with a little python code instead

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import math

def mod(num, a): 
    num = str(num)
    res = 0
    for i in range(0, len(num)): 
        res = (res * 10 + int(num[i])) % a; 
    return res 

f2018 = math.factorial(2018)
primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997]
for x in primes:
  m3 = mod(f2018,math.pow(x,3))
  m4 = mod(f2018,math.pow(x,4))
  if m3==0 and m4!=0 :
    print(x);exit();

#prints: 509