The sum of two numbers is 60 . Each number is smaller than 40 and has exactly six factors. Their common factors are 1,2 and 4 . What are these two numbers?

Let the two numbers be \(a\) and \(b\) with \[ a+b=60, \] and both \(a\) and \(b\) are less than 40 and have exactly six factors. A number has exactly six factors if it is of one of the following forms: 1. \(p^5\), where \(p\) is a prime (giving \(5+1=6\) divisors), 2. \(p^2q\), where \(p\) and \(q\) are distinct primes (giving \((2+1)(1+1)=6\) divisors). Since both numbers are less than 40, the candidates from these forms are limited. 1. The form \(p^5\): - For \(p=2\), we get \(2^5=32\), which is less than 40. - For any larger prime, \(p^5\) will exceed 40. 2. The form \(p^2q\): - For \(p=2\): - \(2^2 \times 3=12\), - \(2^2 \times 5=20\), - \(2^2 \times 7=28\). - For \(p=3\): - \(3^2 \times 2=18\). - (Any other choice yields numbers \(\ge40\) or duplicates.) Thus, the possible numbers with exactly six factors and less than 40 are: \[ 12,\; 18,\; 20,\; 28,\; 32. \] Next, we must choose two numbers from this list such that their sum is 60: - Checking possible pairs, we find: \[ 28+32=60. \] Finally, we verify their common factors. The factor lists are: - Factors of \(28\): \(1,\,2,\,4,\,7,\,14,\,28\), - Factors of \(32\): \(1,\,2,\,4,\,8,\,16,\,32\). Their common factors are: \[ 1,\; 2,\; 4. \] This meets the condition that the numbers have exactly the common factors \(1\), \(2\), and \(4\). Thus, the two numbers are \(28\) and \(32\).