Fermat's Little Theorem states that if "p" is a prime number and "a" is an integer not divisible by "p," then:

a^(p-1) ≡ 1 (mod p)

Let's use Fermat's theorem to find a number "x" such that x⁷³ ≡ 4 (mod 37):

Since 37 is a prime number, we can use Fermat's Little Theorem with "a = x" and "p = 37":

x^(37-1) ≡ 1 (mod 37)

x³⁶ ≡ 1 (mod 37)

Now, rearrange the congruence equation to fit our requirement:

x⁷³ = x^{(2 * 36 + 1)}

Using the property of modular arithmetic:

x^{(2 * 36 + 1)} ≡ (x³⁶)² * x ≡ 1² * x ≡ x (mod 37)

So,  need to find a number "x" such that "x" is congruent to 4 (mod 37). A straightforward solution is x = 4. Let's verify:

4⁷³ ≡ 4 (mod 37)

Now, let's calculate (4⁷³) % 37:

(4⁷³) % 37 = 4

Therefore, x = 4 is the number between 0 and 37 that satisfies x⁷³ ≡ 4 (mod 37) using Fermat's theorem.