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Problem

Find the smallest number, greater than 1, which has a remainder of 1 when divided by any of the numbers 2, 3, 4, 5, 6, or 7.

Solution

We note that 23 = 6 is the smallest number that divides evenly by 2 or 3. In the same way it should be clear that 235 = 30 will be the smallest number that divides by 2, 3, 5, or 6. Although it is obvious that divisibility by 7 requires a factor of 7, to ensure that it divides by 4 = 22, we only need add one extra factor of 2. Hence 23527 = 420 is the smallest number that evenly divides by 2, 3, 4, 5, 6, or 7, and so 421 is the smallest number with a remainder of 1.

What is the smallest number which, when divided by any of the number from 2 to 10, has a remainder of 1?
What about the smallest number which when divided by 2 has a remainder of 1, when divided by 3 has a remainder of 2, or when divided by 4 has a remainder of 3?