I had recently solved a problem.
A number 47_ _74 is multiple of at least two consecutive numbers. Find the numbers. The list of numbers may be of any length $\ge 2$.
I first saw that if they were multiples of 4 numbers then it must be divisible by 4 but it isn't so they are multiples of 2 or 3 numbers. Also all the two or three numbers must be 2-digit or 3-digit. I tried pairing consecutive numbers but no two consecutive numbers produced a result whose units digit was 4. So I tried for 3 number pairs. The only two pairs were $(*2, *3, *4)$ and $(*7, *8, *9)$. (Replace the stars with one-digit numbers). So now since $70\cdot 70\cdot 70=343000 \text{ and } 80\cdot 80\cdot 80 = 512000$. I tried $72\cdot 73\cdot 74$ and $77\cdot 78\cdot 79$ and $77\cdot 78\cdot 79$ produced a result of 474474 and fulfilled the result.
I want to know if my approach is practical. Is it correct? Can you suggest a better way of tackling this problem? I would love new answers. Can you suggest some 'elegant' proof?