Question

28 integers are chosen from the interval [104, 208]. Show that there exit two of them having a common prime divisor.

Answer 1

No. of primes between 104 to 208 => 19

We have so many Integers which are →

divisible by 2 → 206 (2×103)................106(2×53)

divisible by 3 → 183(3×61)................111(3×37)

divisible by 5 → 105,110 ................205

divisible by 7 → 105................203

divisible by 11 → 110................198

divisible by 13 → 169(13×13)................143(13×11)

divisible by 17 → 119 (17×7)................187(17×11)

So that 19 primes and one from every 2,3,5,7,11 and 13 we have total =>19 + 6 => 25,

Such Numbers that doesn’t have any common prime factor.

But, if we take 28 Numbers so their will be 2 such Numbers that will have 1 same prime factor.