I need a number which divides by 2,3,4,5,6 and always leaves remainder 1 but divides by 7 and leaves no remainder

https://www.cut-the-knot.org/blue/chinese.shtml

First find a number that divides 2,3,4,5,and 6 with no remainder. If you need to, you can use the prime factorizations of each of those numbers to figure it out, but by examination, it's clear that our number is 60. If you add 1 to get 61, this divides 2,3,4,5,and 6 with a remainder of 1. Then, just multiply 61 by 7 and your number is 427 (there are an infinite number of solutions to this problem, this is just one of them.)

hopes this helps :)