A8. The sum of the numbers 1 to 123 is 7626 . One number is omitted so that the sum is now a multiple of 4 . How many different numbers could be omitted?
Solution
31
7626 is two more than a multiple of four, which means for the reduced sum to be a multiple of four then the number omitted must also be two more than a multiple of four. The sequence of relevant numbers that could be removed is $2,6,10, \ldots, 122$. If we add two to each of these and then divide by four we get the sequence $1,2,3, \ldots, 31$. This shows that we could omit 31 different numbers.