Notice that each prime block shows up on the board at equal intervals.

White '2' blocks appear every 2 squares.
Red '3' blocks appear every 3 squares.
Orange '5' blocks appear every 5 squares.
Yellow '7' blocks appear every 7 squares.
etc.

Combinations of blocks also appear at predictable intervals.

Numbers with at least two white '2' blocks occur every 4 squares.
Numbers with at least one white '2' block and one red '3' block occur every 6 squares.
Numbers with at least three white '2' blocks occur every 8 squares.
Numbers with at least two red '3' blocks occur every 9 squares.
Numbers with at least one white '2' block and one orange '5' block occur every 10 squares.
etc.

Related to the previous two items, notice that neighboring numbers (those that differ by 1) never have any common prime factors - meaning that they have no common factor other than 1.

Again, related to the previous items, the numbers 17 and 19 often occur only once on the board since any numbers with these factors are separated by 17 and 19 squares respectively. The numbers 11 and 13 also do not occur too frequently.

Use divisibility tests.

For prime numbers...

A number is divisible by 2 if and only if its final digit is even (0, 2, 4, 6, or 8).
A number is divisible by 3 if and only if the sum of its digits is divisible by 3.
A number is divisible by 5 if and only if the final digit is 5 or 0.
To determine if a number is divisible by 7, subtract twice the last digit from the number formed by the remaining digits. The original number is divisible by 7 if and only if this difference is divisible by 7.
To determine if a number is divisible by 11, subtract the sum of the odd-position digits from the sum of the even-position digits. The original number is divisible if and only if this difference is divisible by 11.

For composite numbers...

A number is divisible by 4 if and only if its final two digits are divisible by 4.
A number is divisible by 6 if and only if it is divisible by both 2 and 3.
A number is divisible by 9 if and only if the sum of the it digits is divisible by 9.
A number is divisible by 10 if and only if its final digit is 0.

Use estimation.

To determine if a number is prime, you only need to test prime numbers less than its square root as possible factors.

Pay attention to the final digit in the factors and product. The final digit of the product of the final digits of the factors is equal to the final digit of the product of the original numbers. (Read this a few times. It's not as complicated as it sounds. It might even be something you already know!)

Once you get to know smaller prime numbers, use them to factor larger numbers into primes. For example, once you remember that 53 is prime, you can quickly see that the prime factorization of 106 is 2 ⋅ 53.

If the number is even, take half of it. If it is still even, take half again. This will tell you how many white blocks there are. More importantly, it will help you find the remaining factors.

Break a number apart into factors that you know. For example, if you know that 144 = 12 ⋅ 12, and you know that 12 = 2 ⋅ 2 ⋅ 3, you can see that 144 has four factors of 2 and two factors of 3.

Use completed (or other known) factorizations to help you find new ones. For example, since 120 = 20 ⋅ 6, then 19 ⋅ 6 must be 6 less than 120, or 114. Therefore, 114 has a brown '19' block - and to be more specific, 114 = 19 ⋅ 3 ⋅ 2!

As you play more often, work on getting to know some new facts, such as perfect squares:

11 ⋅ 11 = 121
12 ⋅ 12 = 144
13 ⋅ 13 = 169
14 ⋅ 14 = 196
15 ⋅ 15 = 225
16 ⋅ 16 = 156
17 ⋅ 17 = 289
18 ⋅ 18 = 324
19 ⋅ 19 = 361
etc.