A whole number leaves a remainder of 63 when it is divided by 899. What remainder does the same number leave when it is divided by 29?

Write the number as $899k+63$. Now $899=29\times 31$, so 899$k$ divides exactly by 29 and contributes nothing to the remainder; everything depends on the 63. Since $63=29\times 2+5$, the remainder on division by 29 is 5. The traps: 63 repeats the remainder that was handed to you, but a remainder must always be smaller than the divisor, and 63 is larger than 29; 2 is the quotient of 63 ÷ 29 rather than its remainder; 34 is 63 − 29, one subtraction short of finishing.