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Let P represent the set of all natural numbers from 1 to 100, each of which differs from the sum of its factors by 1, and let Q represent the set of all natural numbers from 1 to 100, each of which is equal to the number of its factors. The L.C.M. of the greatest element of P and the greatest element of Q is

Question

Let P represent the set of all natural numbers from 1 to 100, each of which differs from the sum of its factors by 1, and let Q represent the set of all natural numbers from 1 to 100, each of which is equal to the number of its factors. The L.C.M. of the greatest element of P and the greatest element of Q is

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Solution

To solve this problem, we first need to find the greatest elements of sets P and Q.

  1. Set P: The set of all natural numbers from 1 to 100, each of which differs from the sum of its factors by 1.

The factors of a number are the numbers that divide it exactly. For example, the factors of 6 are 1, 2, 3, and 6. The sum of these factors is 12. But 6 does not differ from 12 by 1, so 6 is not in set P.

By checking the numbers from 1 to 100, we find that the greatest number that fits this condition is 2. The factors of 2 are 1 and 2, and the sum of these factors is 3, which differs from 2 by 1.

  1. Set Q: The set of all natural numbers from 1 to 100, each of which is equal to the number of its factors.

Again, by checking the numbers from 1 to 100, we find that the greatest number that fits this condition is 1. The only factor of 1 is 1 itself.

  1. The least common multiple (L.C.M.) of the greatest elements of P and Q:

The L.C.M. of 2 and 1 is 2.

So, the L.C.M. of the greatest element of P and the greatest element of Q is 2.

This problem has been solved

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