representation as a sum of smaller divisors. For this to be true, the first prime must equal 2 and, for every from 2 to , each successive prime must obey the inequality where denotes the sum of the divisors of x. For example, 2 × 32 × 29 × 823 = 429606 is practical, because the inequality above holds for each of its prime factors: 3 ≤ σ(2) + 1 = 4, 29 ≤ σ(2 × 32) + 1 = 40, and 823 ≤ σ(2 × 32 × 29) + 1 = 1171. The condition stated above is necessary and sufficient for a number to be practical. In one direction, this condition is necessary in order to be able to represent as a sum of divisors of , because if the inequality failed to be true then even adding together all the smaller divisors would give a sum too small to reach . In the other direction, the condition is sufficient, as can be shown by induction. More strongly, if the factorization of satisfies the condition above, then any can be represented as a sum of divisors of , by the following sequence of steps: By induction on , it can be shown that . Hence . Since the internals cover for , there are such a and some such that . Since and can be shown by induction to be practical, we can find a representation of q as a sum of divisors of . Since , and since can be shown by induction to be practical, we can find a representation of r as a sum of divisors of . The divisors representing r, together with times each of the divisors representing q, together form a representation of m as a sum of divisors of . Properties The only odd practical number is 1, because if is an odd number greater than 2, then 2 cannot be expressed as the sum of distinct divisors More strongly, observes that other than 1 and 2, every practical number is divisible by 4 or 6 (or both). The product of two practical numbers is also a practical number. More strongly the least common multiple of any two practical numbers is also a practical number. Equivalently, the set of all practical numbers is closed under multiplication. From the above characterization by Stewart and Sierpiński it can be seen that if is a