In this case, nothing that I know of. Again, that's the kind of math that is classified as "number theory". It's just pure investigation of relationships between numbers. Occasionally you gain some insight that is useful.
A lot of cryptography used to be just number theory until computers came along and were powerful enough to make use of it. How to tell if something if someting is divisible by 3. Checksums as used on credit cards. Euler's algorighm.
No, sexy primes don't really have a point other than they are identified and there is probably some unused conjecture that they are infinite in number.
Using Euclid's algorithm (not Euler's) is certainly not the easiest way of checking for divisibility by 3 - a number is divisible by 3 if and only if it's sum of digits is divisible by 3. You can repeat the process until you have one digit.
Or you can form the alternating sum of the bits, e.g. for 0b10011001 you calculate 1-0+0-1+1-0+0-1 = 0 which is divisible by three. (That's similar to the divisibility test by 11 of a number in base-10, or more generally testing if a number in base `b` is divisible by b+1)
A lot of cryptography used to be just number theory until computers came along and were powerful enough to make use of it. How to tell if something if someting is divisible by 3. Checksums as used on credit cards. Euler's algorighm.
No, sexy primes don't really have a point other than they are identified and there is probably some unused conjecture that they are infinite in number.