My grandparents have four grandchildren. The product of the ages of the four grandchildren is 67 184. The youngest grandchild is younger than 10, and is also 30 years younger than the oldest grandchild. Determine all possibilities for the ages of my grandparents’ grandchildren.

The possible ages of the four grandchildren are a = 4, b = 19, c = 26, and d = 34

Step-by-step explanation:

The given parameters are;

The number of grandchildren in the family = 4

The product of the ages of the four grand children = 67184

The age of the youngest grandchild < 10

The age of the oldest grandchild = 30 + The age of the youngest grandchild

Let a represent the age of the youngest grandchild, and let b, and c represent the ages of the other two intermediate grandchild

Therefore, we have;

a < 10

The age of the oldest grandchild = a + 30 < 10 + 30

∴ The age of the oldest grandchild < 40

The product of the ages of the four grandchildren = a × b × c × (a + 30) = 67184

The factors of 67184 that are between 1 and 40 are;

1, 2, 4, 8, 13, 16, 17, 19, 26, 34, 38

Taking a = 8, we have;

The age of the oldest grandchild × The age of the youngest grandchild = a × (a + 30) = 8 × 38 = 342

Therefore. a × b = 67184/(a × (a + 30) = 196.44

Therefore, a ≠ 8

For a = 4, we have the age of the oldest grandchild = a + 30 =  4 + 30 = 34

The age of the oldest grandchild × The age of the youngest grandchild = a × (a + 30) = 4 × 34 = 136

Therefore. a × b = 67184/(a × (a + 30) = 494

We find that the other factors of 67184, which are 19 and 26 have a product of 494

Therefore, the possible ages of the four grandchildren are a = 4, b = 19, c = 26, and d = 34

To give, 4 × 19 × 26 × 34 = 67,184.

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