What is the smallest positive integer which gives 1 as a remainder when divided by the numbers 2,3,4,5,6

To find the smallest positive integer that gives a remainder of 1 when divided by the numbers 2, 3, 4, 5, and 6, you can follow these steps:

Step 1: Start by finding the least common multiple (LCM) of the numbers 2, 3, 4, 5, and 6. The LCM is the smallest number that is divisible by all of these numbers.

To do this, list the multiples of each number until you find a common multiple.

Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ...
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ...
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ...
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, ...
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, ...

From these lists, we can see that the LCM of 2, 3, 4, 5, and 6 is 60.

Step 2: Now, we need to find the smallest positive integer that gives a remainder of 1 when divided by 60.

We can start by listing out some multiples of 60 and checking if they satisfy the conditions:

60 * 1 = 60 (remainder: 0)
60 * 2 = 120 (remainder: 0)
60 * 3 = 180 (remainder: 0)
60 * 4 = 240 (remainder: 0)
60 * 5 = 300 (remainder: 0)
60 * 6 = 360 (remainder: 0)
60 * 7 = 420 (remainder: 0)
60 * 8 = 480 (remainder: 0)
60 * 9 = 540 (remainder: 0)
60 * 10 = 600 (remainder: 0)

As we can see, none of the multiples of 60 give a remainder of 1 when divided by 60.

Step 3: To find the smallest positive integer that gives a remainder of 1 when divided by 60, we need to start from 1 and increment until we find a number that satisfies the condition.

Let's try this incrementally:

1 ÷ 60 = 0 (remainder: 1)
2 ÷ 60 = 0 (remainder: 2)
3 ÷ 60 = 0 (remainder: 3)
...
59 ÷ 60 = 0 (remainder: 59)
60 ÷ 60 = 1 (remainder: 0)
61 ÷ 60 = 1 (remainder: 1)

Therefore, the smallest positive integer that gives a remainder of 1 when divided by 2, 3, 4, 5, and 6 is 61.

Note: Another way to solve this is by finding the smallest positive integer that is one less than a multiple of the LCM of the numbers. In this case, the LCM is 60, so we look for a number that is one less than a multiple of 60, which is 59. Adding 1 to 59 gives us the answer of 60.

The LCM is 60

So how about 61 as the lowest number.

or
for the 2 it would be 3,5,7,9,11,13,....,57,59,61,63,..
for the 3 it would be 4,7,10,13,16,....,58,61,64,...
for the 4 it would be 5,9,13,17,...., 49,53,57,61,65,...
for the 5 it would be 6,11,16,....., 51,56,61,66,....
for the 6 it would be 7, 13,19,...., 49,55,61,67,...

For a while I was cheering for 13, but it looks like 61 is the winner.