What is the smallest whole number that can be divided exactly by every one of 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10?

Take the highest power of each prime that appears anywhere in the list: $8=2^3$ forces $2^3,9=3^2$ forces $3^2$, and 5 and 7 each appear once. That gives $2^3\times 3^2\times 5\times 7=2520$, and 2520 really does divide by all ten. The near-miss $1260=2^2\times 3^2\times 5\times 7$ uses only $2^2$ and so fails on 8, since 1260 ÷ 8 = 157.5. The other two traps multiply far too much: 5040 = 7! and 3628800 = 10!, both multiples of 2520 but nowhere near the smallest — multiplying every number together is exactly what the LCM is designed to avoid.