let x be the smallest composite number that has no prime factors less than 10

x also has no factors less than 10 (since every composite factor less than 10 is a product of prime factors less than 10) thus the first possible factor of x could be 10

since 10 = 2*5 follows 2 and 5 are factors of x a,d 2<10, 5<10 which is not allowed thus 10 is not factor of x

the next number is 11 < 10 and we have no problem with 11 thus 11 can be a factor of x

thus the number 11*11= 121 is a composite number that has not prime factors less than 10

lets prove that 121 is the smallest such number

let y is composite number that has no prime factors less than 10 and y < x

y < 11*11 => y has a factor smaller than 11 but no less than 10 => 10 is a factor of y

10 = 2*5 => 2 and 5 are factors of y we get contradiction

121 is the smallest composite number that has no prime factors less than 10.