What numbers are between 64 and 77 that the sum of their digits is prime and they have more than three factors?

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mwmovr40 eNotes educator| Certified Educator

To help with this problem we must know what it means to be a prime number and what it means to have more than three factors.

A prime number is a number that can have only two numbers that divide into it evenly.  One of the two numbers is always the number 1, and the other is the number itself.  For example, 7 is prime because their are only two numbers that divide into it evenly: 1 and 7.

We should remember that any even number (except 2 itself) can not be prime because they would have at least three numbers that can divide into them evenly.  4 for example is even and can not be prime because it can be divided equally by 1, 2, and 4.  We should also remember that if you add two even numbers together you will always get an even answer and if you add two odd numbers together you will get an even answer.

So, we need to look for all of the numbers between 64 and 77 that when you add their digits together make a prime number.  We can throw out any number that has an even digit for the ones place because between 64 and 70 because they will add up to an even answer and can't be prime.  We can throw out all the numbers from 71 to 77 that have an odd number in the ones place for the same reason.  That just leaves 65, 67, 69, 72, 74, and 76.  From these we can see which ones have sums that are prime:

6+5 = 11 prime

6+7 = 13 prime

6+9 = 15 not prime (15 has factors 1, 3, 5, and 15)

7 + 2 = 9 not prime (9 has factors 1, 3, and 9)

7 + 4 = 11 prime

7 + 6 = 13

Now we can look 65, 67, 74, and 76 and use the other rule: which ones have more than three factors.  Factors are the numbers you add together to make the number you start with. So

65 has 4 factors of 1 and 65, 5 and 13

67 is prime

74 has 4 factors of 1 and 74, 2 and 37

and 76 has factors of 1 and 76, 2 and 38, 4 and 19

So 65, 74, and 76 are the numbers between 64 and 77 which have digits which sum to prime numbers and which themselves have more than three factors each.