Via prime factorization:
[tex]100=10^2=2^2\cdot5^2[/tex]
[tex]254=2\cdot127[/tex]
of which the only common factor is a single power of 2, so [tex]\mathrm{gcd}(100,254)=2[/tex].
Via the Euclidean algorithm:
[tex]254=2\cdot100+54[/tex]
[tex]100=1\cdot54+46[/tex]
[tex]54=1\cdot46+8[/tex]
[tex]46=5\cdot8+6[/tex]
[tex]8=1\cdot6+2[/tex]
[tex]6=3\cdot2+0[/tex]
which means [tex]\mathrm{gcd}(100,254)=2[/tex], as expected.
