What is the smallest integer under certain conditions ?

What is the smallest integer greater than 2 that leaves a remainder of 2 when divided by 3,4,5 or 6?

Let x be the required smallest integer. According to the conditions given,

3q₁ + 2 = 4q₂ + 2 = 5q₃ + 2 = 6q₄ + 2= x

Or 3q₁ = 4q₂= 5q₃ = 6q₄ = x-2

It means that x-2 is divisible by the divisors 3,4,5 and 6.. The smallest number to which 3,4,5 and 6 are all perfect divisors can be determined by multiplying all the common factors of the divisors.

3[ 3,4,5,6

2[ 1,4,5,2

   1,2,5,1

The smallest integer x -2 to which 3,4,5 and 6 are perfect divisors is given by 3x2x2x5 = 60;or x = 62.

Variant-1

Condition                                        Answer                              explanation

Divisors 3,4                                      14                               3x4+2

               3,4,5                                   62                               3x4x5+2

              3,4,5,6                                62                                2x5x6+2

              3,4,5,6,7                           422                               3x4x5x7+2

              3,4,5,6,7,8                       842                               3x5x7x8+2

              3,4,5,6,7,8,9                  2522                              5x7x8x9 +2

Variant-2

Find the smallest integer (x) greater than 3 that leaves a remainder when divided by 4,5,6,7.

4q₁ +3 = 5q₂ + 3 = 6q₃ + 3 = 7q₄ + 3 =  x

 Or 4q₁ = 5q₂ = 6q₃ = 7q₄ = x – 3

The smallest x-3 is given by the product of all common factors to the divisors 4,5,6,7 that is 3x4x5x7 = 420 or x = 423

 Condition               solution         explanation

 Divisors 4,5               23                   4x5+3

                4,5,6            63                  3x4x5+3

               4,5,6,7       423                  3x4x5x7+3

              4x5x6,7,8   843                 2x3x4x5x7+3

             4,5,6,7,8,9  2523              2x4x5x7x8 + 3