Three of these four numbers share a factor that the fourth one does not: 49, 63, 77, 87. Which is the odd number out?

Split each number up: $49=7\times 7,63=7\times 9$ and $77=7\times 11$, so three of them are multiples of 7. But $87=3\times 29$, and 87 ÷ 7 leaves a remainder of 3, so 87 is the one that breaks the pattern. The trap is 49: it stands out to the eye because it is the only perfect square in the list, but being a square does not stop it being a multiple of 7, and a square/non-square split would put one number against three for the wrong reason.