There is probably a better way to do this but:
since it leave a remainder of 1 when divided by 2, it is an odd number
the odd multiples of 7 are
7,
21,35,49,63,77
91,105,119,133,147,
161,175, 189, etc
you want one more than a multiple of 5, so it is number ending in 1
answer is 301
the least common multiple of 2,3,4,5,6 is 60
you want a multiple of 60 plus 1 that is also a multiple of 7:301