What is the greatest four-digit number that is a multiple of 88?

Divide the largest four-digit number by 88: 9999 ÷ 88 = 113 remainder 55, so the answer is $88\times 113=9944$. Watch out for 9856 — it really is a multiple $(88\times 112)$, just not the greatest one, so it catches anyone who stops one step early. 9968 is not a multiple at all, and $9999=88\times 113+55$.