# Sale of cattle Ashraf and Ali were two Egyptian camel drivers from Sharkieh province. One day they decided to change jobs and work on grazing, so they moved to Birqahs camel market to sell their animals. They sold each animal for an amount of pounds equal to the number of camels they sold. With that money they bought as many sheep as they could at a rate of 10 pounds each and with the remaining money they acquired a goat.

On the way back they discussed and decided to separate. When they made the distribution of the sheep, they saw that there was one left over for what Ashraf said: "I keep the sheep and you take the goat," to which he replied: "It's not fair, the goat is worth less." Okay, said Ashraf, "I'll give you one of my dogs, too."

What is the price of the dog?

Extracted from the page of the Labor University of Gijón.

#### Solution

The amount of pounds they got from the sale of their camels is a perfect square since each camel was sold for an amount equal to the total camels.

Obviously, the number of sheep they bought was odd, indicating that the number of tens of the amount of money was odd and the number of the units was not 0 since with that amount they bought the goat.

At this point we need to know the properties of perfect square numbers. A square number can be terminated in the digits 00, 1, 4, 6, 9, or 25, as follows:

• If the last digit of a number is 0, its square ends in 00 and the preceding digits must also be a square.
• If the last digit of a number is 1 or 9, its square ends in 1 and the number formed by its precedent must be divisible by four. The tens digit is even.
• If the last digit of a number is 2 or 8, its square ends in 4 and the preceding digit must be an even number. The tens digit is even.
• If the last digit of a number is 3 or 7, its square ends in the digit 9 and the number formed by its previous digits must be divisible by four. The tens digit is even.
• If the last digit of a number is 4 or 6, its square ends in 6 and the tens digit is odd.
• If the last digit of a number is 5, its square ends in 25 and the preceding digits must be 0, 2, 06, or 56.

Looking at the previous properties, it follows that the amount of money obtained from the sale of camels is a perfect square of a number ending in 4 or 6. The number of units is 6 which is the price of the goat. The tens digit is odd, which indicates that, in effect, the number of sheep was odd (its price was 10 pounds). So that:

1 sheep = 1 goat + 1 dog
10 pounds = 6 pounds + 1 dog

So we deduce that the dog cost 4 pounds.