Here's another way.

The number 2×2×3×5 = 60 is divisible by 2,3,4,5, and 6.

So 1 less than the number we're looking for has all these factors,
and so is a multiple of 60.

So we are looking for a number that is one more than a multiple of
60 that is also a multiple of 7.

Write down the first few multiples of 60:

60, 120, 180, 240, 300, 360, 420

Add 1 to each

61, 121, 181, 241, 301, 361, 421

Divide each by 7.  You'll get a decimal each time until you come to
301.  When you divide 301÷7 and you get 43

So the answer is 301.

Edwin