Prove that for any 9 consecutive natural numbers, there exists one number that is prime to the other 8 numbers.

Prove: a number is even  iff its last digit is even.

identify all natural numbers having exactly 28 divisors.

Denote by an the nth positive integer that is not a perfect square. For example, we have the set of non-perfect-square integers {2,3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17,...}, and then a1 = 2, a2 = 3, az = 5, a4 = 6, 7, 8, 10, ag = 11,... a5 = a6 a7 || Show that an = n+ {/n} where {x} denotes the integer closest to the real number x.