Question

24, Find the sum of all numbers less than 1000 which are exactly divisible
by 2, 3, 4, 5, 6 and 7.​

Answer 1

The relation between multiples and factors is the following:

• $\sf{multiple\:=\:factor\:\times\:factor}$

If numbers are divisible by 2, 3, 4, 5, 6, and 7, it must have factors. Since 'and' means satisfying all at the same time, there needs to be 2, 3, 4, 5, 6, and 7 at the same time as factors. In this case, LCM(Lowest Common Factor) is used.

The factor needs to be the LCM of 2, 3, 4, 5, 6, and 7.

$\[\begin{array}{@{}l|l@{}}2 & 2\:3\:4\:5\:6\:7\\ \cline{2-2} 3 & 1\:3\:2\:5\:3\:7\\ \cline{2-2}\multicolumn{1}{c}{} & 1\:1\:2\:5\:1\:7\end{array}\]$

→ $2\times3\times2\times5\times7$

$=2^2\times3\times5\times7$

$=20\times21$

$=420$

The number must be the following:

• $\sf{multiple\:=420\:\times\:factor}$

Possible numbers are:

• 420 × 1 = 420
• 420 × 2 = 840

∴The sum is equal to 420 × 3 = 1260.