Simple Number Theory: "Mr. White is an approximately forty years old father..."

[Steven G]

Simple Number Theory: "Mr. White is an approximately forty years old father..."

Mr. White is an approximately forty years old father with 4 sons of distinct ages. Writing his age 3 times in succession, we get a 6-digit number that is equal to the product of his age, his wife's age and his 4 sons' ages.
Give the sum of his wife's age and all 4 sons' ages.

 

[ksdhart2]

Suppose the White family has the following ages:

• Mr. White: 39
• Mrs. White: 37
• Son 1: 13
• Son 2: 7
• Son 3: 3
• Son 4: 1

Then we have \(\displaystyle 393939 = 39 \cdot 37 \cdot 13 \cdot 7 \cdot 3 \cdot 1\), as required. More generally, fixing the wife and kids' ages as above, Mr. White can be anywhere between the ages of 35 and 44 (I figure 34 is "approximately 30" and 45 is "approximately 50"), and it still works. Because for any two-digit age of the form xy, we have \(\displaystyle \dfrac{xyxyxy}{xy} = 10101\), whose prime factors are the ages of Mrs. White and Sons 1, 2, and 3. Then Son 4's age is 1 so as to not change the product.

Edit: Oops. I made a big mistake last night! I have since revisited the problem. I was absolutely wrong. There is no other way for five numbers to multiply together and get 10101. Additionally, I now see that the problem asks for only the sum of the ages of Mrs. White and the four sons, so we don't need to know Mr. White's exact age, so there is, in fact, a unique solution.

 

[Steven G]

Suppose the White family has the following ages:

• Mr. White: 39
• Mrs. White: 37
• Son 1: 13
• Son 2: 7
• Son 3: 3
• Son 4: 1

Then we have \(\displaystyle 393939 = 39 \cdot 37 \cdot 13 \cdot 7 \cdot 3 \cdot 1\), as required. More generally, fixing the wife and kids' ages as above, Mr. White can be anywhere between the ages of 35 and 44 (I figure 34 is "approximately 30" and 45 is "approximately 50"), and it still works. Because for any two-digit age of the form xy, we have \(\displaystyle \dfrac{xyxyxy}{xy} = 10101\), whose prime factors are the ages of Mrs. White and Sons 1, 2, and 3. Then Son 4's age is 1 so as to not change the product.

Edit: Oops. I made a big mistake last night! I have since revisited the problem. I was absolutely wrong. There is no other way for five numbers to multiply together and get 10101. Additionally, I now see that the problem asks for only the sum of the ages of Mrs. White and the four sons, so we don't need to know Mr. White's exact age, so there is, in fact, a unique solution.

Nicely done.