In Their Prime
Jonathan is two years younger than his
wife Hazel. Their current ages are both
prime numbers. Next year Hazel's age will
be a multiple of 11. Jonathan's age will be
the product of two consecutive numbers.
Question: How old are Jonathan and Hazel? Use the space below to explain your
response.
Extra: Two prime numbers that differ by two (like Jonathan and Hazel's current ages) are called
twin primes. They are always separated by a single even number. Find all the twin primes less
than 100. Starting with the twin primes 5 and 7, what is the greatest common factor that those
single even numbers all share? Explain how you found your answers.
Prime: can be divided evenly only by 1 or itself and it must be a whole number greater than 1.
Composite: can be divided evenly by numbers other than 1 or itself.
Expected Solution
Hazel is 43 years old and Jonathan is 41 years old. Extra: The greatest common factor that the
even numbers between prime twins above 5 and 7 share is 6.
First I remembered that my Cool Math book had an activity to find all the prime numbers
between 1 and 100. You get a hundreds chart with all the numbers that are NOT prime
numbers crossed out. Then I started to look at the chart to see what prime numbers would fit
the clues.
I saw that both ages were prime numbers and Jonathan was two years older than Hazel. So, I
needed to find two prime numbers that were two apart. Since there was more than one pair of
two prime numbers, I decided to go to the next clue.
I saw that next year, Hazel’s age had to be a multiple of 11. Then this year, her age would be
one year less than that multiple of 11. I did 2 X 11 = 22, and then 22 - 1 = 21, but 21 is not a
prime number. Then I did 3 X 11 = 33, and then 33 - 1 = 32, but 32 is not a prime number.
Then I did 4 X 11 = 44, and then 44 - 1 = 43, and 43 is a prime number.
I decided to make Hazel’s age 43, and see if the rest of the clues would work.
43 - 2 = 41, and 41 is a prime number. So, Jonathan could be 41 years old.
The last clue was that next year, Jonathan’s age will be the product of two consecutive
numbers. If Jonathan was 41 this year, next year he would be 42. 6 X 7 = 42, and 6 and 7 are
consecutive numbers. So the last clue worked. Since all the clues worked, I know I am right.
Extra:
I made a chart of all the twin primes, and the number in between them. I looked at the chart I
used for the first part to find them.
Twin Primes
5-7
11 - 13
17 - 19
29 - 31
41 - 43
59 - 61
71 - 73
Number between
6
12
18
30
42
60
72
I started with 6 and found 1, 6, 2, and 3 for factors. Since 6 is the biggest factor, I started with 6
to see if it is a factor of all the other numbers.
Next is 12, and 6 X 2 = 12. So 6 would work.
Next is 18, and 6 X 3 = 18. So 6 would work.
Next is 30, and 6 X 5 = 30. So 6 would work.
Next is 42, and 6 X 7 = 42. So 6 would work.
Next is 60, and 6 X 10 = 60. So 6 would work.
Next is 72, and 6 X 12 = 72. So 6 would work.
So, the greatest common factor that the even numbers between prime twins above 5 and 7
share is 6.