Question

Show that 2^105 + 3^105 is divisible by 5, 7, 11, 25 but not 13

Answer 1

Given: The term 2^105 + 3^105

To find: Show that 2^105 + 3^105 is divisible by 5, 7, 11, 25 but not 13.

Solution:

• Now we have given the term as:

                    2^105 + 3^105

• We can rewrite it as:

                    2^(3x5x7) + 3^(3x5x7)   ....................(i)

                    8^(5x7) + 27^(5x7)

• Now we know the property:

                    a^m + a^n = a^(m+n)   .......................(ii)

• Applying this, we get:

                    (8 + 27)^(5x7)

                    35^(5x7)

• Now 35 is divisible by 5 and 7. .....................(iii)
• Again consider (i), we have:

                    2^(5x3x7) + 3^(5x3x7)

                    32^(3x7) + 243^(3x7)

• Applying (ii), we get:

                    (32 + 243)^(3x7)

                    275^(3x7)

• Now 275 is divisible by 11 and 25......................(iv)

• Again consider (i), we have:

                    2^(5x3x7) + 3^(5x3x7)

                    128^(5x3) + 2187^(5x3)

• Applying (ii), we get:

                    (128+2187)^(5x3)

                    2315^(5x3)

• Here 13 is not the factor of 2315......................(v)

Answer:

                  So from (iii), (iv) and (v), we proved that 2^105 + 3^105 is divisible by 5, 7, 11, 25 but not 13.