Factor riddles and perfect numbers

Factors help to solve 'Who am I?' riddles. Here is one: 'I am a number less than 40. One of my factors is 7. The sum of my digits is 8.' Go clue by clue:

• 7 is a factor, so the number is in the 7 times table: 7, 14, 21, 28 or 35 (below 40).
• Their digit sums are 7, 5, 3, 10 and 8.
• Only 35 has digit sum 8. So the answer is 35.

Another: 'I am a number less than 100. Two of my factors are 3 and 5. One of my digits is 1 more than the other.' A number with both 3 and 5 hidden inside it is a multiple of 15: 15, 30, 45, 60, 75, 90. Only 45 has one digit (5) that is 1 more than the other (4). So the answer is 45.

The book also tells us about perfect numbers. A number is perfect when the sum of all its factors, including the number itself, is twice the number. 28 is perfect: its factors are 1, 2, 4, 7, 14 and 28, and 1 + 2 + 4 + 7 + 14 + 28 = 56, which is 2 × 28.

Is there a perfect number between 1 and 10? Try 6: its factors are 1, 2, 3 and 6, and 1 + 2 + 3 + 6 = 12 = 2 × 6. So 6 is a perfect number. The others do not work: for 8, 1 + 2 + 4 + 8 = 15, not 16.