## Task 6 - 030716

a) Find the smallest natural number that leaves a remainder of 1 when divided by 2, 3, 4, 5, and 6, but is divisible by 7.

b) Name two more numbers with this property and explain how to generate infinitely many such numbers!
a) The least common multiple of 2, 3, 4, 5, and 6 is 60. Therefore, the numbers 1, 61, 121, 181, ... leave a remainder of 1 when divided by 2, 3, 4, 5, and 6. By trial, it is found that 301 is the smallest of these numbers that is divisible by 7.

b) Other numbers include, for example, 721 and 1141. By adding 420 to such a number, one always obtains another number with the desired property.

Solutions of the I. Round 1963 taken from [5]

### 4.5.2 II. Round 1963, Class 7