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My Grandson - Age Word Problem

by Andrew
(New York)











































My grandson, my son (his father), and I share the same birthday. This year I remarked to
my son, “You know, of course, that today all three of our ages have become prime
numbers. But did you realize that for the last 17 years prior to this year, whenever your
age was a prime number, so was mine, and vice versa?”
“Yes,” he replied. “And 17 years from now, I will say to my son, “You know, for the last
18 years prior to this year, whenever your age was a prime number, so was mine, and
vice versa.”
How old are we?

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Oct 03, 2011
My Grandson - Age Word Problem
by: Staff

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Part II

A repeating pattern of -6, -2, -6 appears from age 53 to 67 (age 67, going back in time). It is:

53 . . . . . -6
59 . . . . . -2
61 . . . . . -6
67 . . . . . xx


The same repeating pattern of -6, -2, -6 appears from age 23 to 37 (age 37, going back in time). It is:

23 . . . . . -6
29 . . . . . -2
31 . . . . . -6
37 . . . . . xx


Therefore, the age of the grandfather today and the son today, are probably:

Grandfather: 67

Father: 37


However, you can’t be sure until the age of the grandson is calculated.



Calculation of the possible ages for the son and grandson:

The son is older than the grandson.

The age of the son and the age of the grandson are both prime numbers this year.

Going forward in time, the pattern for the son and grandson must be the same.


A repeating pattern of -4, -2, -4, -6 appears from age 37, going forward. It is:

37 . . . . . -4
41 . . . . . -2
43 . . . . . -4
47 . . . . . -6

The same repeating pattern of -4, -2, -4, -6 appears from age 13, going forward. It is:

13 . . . . . -4
17 . . . . . -2
19 . . . . . -4
23 . . . . . -6


Therefore, the age of the grandson is 13.


The final answer:

Grandfather: 67

Father: 37

Grandson: 13.




Thanks for writing.

Staff
www.solving-math-problems.com


Oct 03, 2011
My Grandson - Age Word Problem
by: Staff


Part I

Question:

My grandson, my son (his father), and I share the same birthday. This year I remarked to
my son, “You know, of course, that today all three of our ages have become prime
numbers. But did you realize that for the last 17 years prior to this year, whenever your
age was a prime number, so was mine, and vice versa?”
“Yes,” he replied. “And 17 years from now, I will say to my son, “You know, for the last
18 years prior to this year, whenever your age was a prime number, so was mine, and
vice versa.”
How old are we?


Answer:


Read the problem “very” carefully.

The problem can be broken down into two separate problems:

1. The ages of the Grandfather and the Son

2. The ages of the Son and the Grandson

All the prime numbers between 1 and 200 are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199

However, I’m going to consider only those prime numbers under 100 because I do not think the grandfather will be over 97 years old.

These are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97


The grandson, son (his father), and grandfather share the same birthday. This makes the years between them the same, regardless of the time of year.

I’m also going to assume the grandfather is older than the son, and the son is older than the grandson.


The problem stated that today all three of our ages have become prime numbers.




Calculation of the possible ages for the grandfather and son:

The grandfather is older than the son.

The age of the grandfather and the age of the son are both prime numbers this year.

For the last 17 years prior to this year, whenever the grandfather’s age was a prime number, so was the sons, and vice versa.

To find possible ages for the grandfather and son, you can calculate the sequential difference between all the prime numbers, and then look for a repeating pattern.

Prime Number . . . Difference Going Back in Time (current prime minus prior prime year)

2 . . . . . . xx
3 . . . . . . -2
5 . . . . . . -2
7 . . . . . . -4
11 . . . . . -2
13 . . . . . -4
17 . . . . . -2
19 . . . . . -4
23 . . . . . -6
29 . . . . . -2
31 . . . . . -6
37 . . . . . -4
41 . . . . . -2
43 . . . . . -4
47 . . . . . -6
53 . . . . . -6
59 . . . . . -2
61 . . . . . -6
67 . . . . . -4
71 . . . . . -2
73 . . . . . . -6
79 . . . . . . -4
83 . . . . . .-6
89 . . . . . . -8
97 . . . . . . xx

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