Yes.

The further you study math, the fewer numerical calculations you have to do. After you get a solid grasp of how to compute by hand, it becomes less important how you arrive at the numerical answer. Concerning fractions, I usually repeatedly divide the numerator and the denominator by the first suitable number I can find.

3990 and 6762 are divisible by 6 because they are even and the sum of their digits is divisible by 3. For 7, they probably tried it and found that it divides both numbers. E.g., 665 = 630 + 35 = 7 * 90 + 7 * 5 and 1127 = 1120 + 7 = 7 * 16 + 7 * 1.

For really big numbers, Euclid's algorithm is more efficient because there are no currently known efficient factorization algorithms. "When the numbers are very large, no efficient, non-quantum integer factorization algorithm is known; an effort concluded in 2009 by several researchers factored a 232-digit number (RSA-768), utilizing hundreds of machines over a span of 2 years. The presumed difficulty of this problem is at the heart of widely used algorithms in cryptography such as RSA" (Wikipedia).