Question 3 - Figure it out - Page 110, 111 - Chapter 5 Class 6 - Prime Time (Ganita Prakash)

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Question 3 A number for which the sum of all its factors is equal to twice the number is called a perfect number. The number 28 is a perfect number. Its factors are 1, 2, 4, 7, 14 and 28. Their sum is 56 which is twice 28. Find a perfect number between 1 and 10.If Sum of all factors is equal to twice the number, then number is called a perfect number. Since we need to find a perfect number between 1 and 10, we will check for numbers 2, 3, 5, 6, 7, 8, 9 For 2 Factors of 2 are 1, 2 Sum of factors = 1 + 2 = 3 Since 3 is not twice of 2 ∴ Sum of factors is equal to twice the number So, 2 is not a perfect number For 3 Factors of 3 are 1, 3 Sum of factors = 1 + 3 = 4 Since 4 is not twice of 3 ∴ Sum of factors is equal to twice the number So, 3 is not a perfect number For 4 Factors of 4 are 1, 2, 4 Sum of factors = 1 + 2 + 4 = 7 Since 7 is not twice of 4 ∴ Sum of factors is equal to twice the number So, 4 is not a perfect number For 5 Factors of 5 are 1, 5 Sum of factors = 1 + 3 = 4 Since 4 is not twice of 5 ∴ Sum of factors is equal to twice the number So, 5 is not a perfect number For 6 Factors of 6 are 1, 2, 3, 6 Sum of factors = 1 + 2 + 3 + 6 = 12 Since 12 = 2 × 6 ∴ Sum of factors is equal to twice the number So, 6 is a perfect number For 7 Factors of 7 are 1, 7 Sum of factors = 1 + 7 = 8 Since 8 is not twice of 7 ∴ Sum of factors is equal to twice the number So, 7 is not a perfect number For 8 Factors of 8 are 1, 2, 4, 8 Sum of factors = 1 + 2 + 4 + 8 = 15 Since 15 is not twice of 8 ∴ Sum of factors is equal to twice the number So, 8 is not a perfect number For 9 Factors of 9 are 1, 3, 9 Sum of factors = 1 + 3 + 9 = 13 Since 13 is not twice of 9 ∴ Sum of factors is equal to twice the number So, 0 is not a perfect number Thus, only 6 is a perfect number between 1 and 100