### Brain Teasers

# Rex and Ralph: Daughter Multiplication

Two mathematicians, Rex and Ralph, have an ongoing competition to stump each other. Rex thinks he may finally get Ralph with his latest. Rex tells Ralph that he has four daughters, ages seven to twenty-three.

"Multiply each of my daughter's age by each of her three sisters' ages. For each daughter, take the product of the three resulting numbers. Then take the product of these four results. Divide this total by the product of the four daughter's ages and discard the fractional portion. Next subtract the product of the four daughter's ages. Now divide this value by 10."

"What is the remainder?" Rex asks.

Ralph thinks about this for about a minute before answering.

Rex is disappointed that he has once again failed to stump Ralph.

What was Ralph's answer?

"Multiply each of my daughter's age by each of her three sisters' ages. For each daughter, take the product of the three resulting numbers. Then take the product of these four results. Divide this total by the product of the four daughter's ages and discard the fractional portion. Next subtract the product of the four daughter's ages. Now divide this value by 10."

"What is the remainder?" Rex asks.

Ralph thinks about this for about a minute before answering.

Rex is disappointed that he has once again failed to stump Ralph.

What was Ralph's answer?

### Hint

There will be no fractional portion to discard.Oh, and the ages of Rex's daughters are irrelevant.

### Answer

Zero.Choose four positive integers, A, B, C and D.

Step 1 yields:

A: AB, AC, AD

B: BA, BC, BD

C: CA, CB, CD

D: DA, DB, DC

Step 2 yields:

A: (A^3)BCD

B: (B^3)ACD

C: (C^3)ABD

D: (D^3)ABC

Step 3 yields:

(ABCD)^6

Step 4 yields:

((ABCD)^6)/(ABCD) = (ABCD)^5

For any positive integer x, x mod 10 = (x^5) mod 10. Therefore,

(ABCD)^5 mod 10 = (ABCD) mod 10

and

((ABCD)^5 - (ABCD)) mod 10 = 0

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