How many camels

Question

A camel shepherd bought a caravan of camels. On his journey through the desert, he noticed that the camels all had a very bad disease and would be of no use for traveling any further. He returned to the seller and asked for new camels. The seller said, “I will gladly give you the same number of camels that you bought if you can tell me how many camels you started with.” The camel shepherd could not remember exactly how many he had bought, but he knew it was less than 100. He remembered that if the camels walked in rows of 2, one walked alone; in rows of 3, one walked alone; in rows of 4, one waked alone; and in rows of 5, one walked alone. How many camels were in the caravan?

Answer 1

The number of camels divided by 2,3,4, or 5 will have 1 remainder. Since 4=2*2, the information about rows of two is not a new information, it is redundant. Every number that has a 1 remainder when divided by 4, will also have 1 remainder when divided by 2. So we are looking for a number less than 100 that has a remainder of 1 when divide by 3 or 4 or 5.

If the remainder when divide by 3 or 4 is 1, then the remainder when divided by 3*4=12 will also be 1.

If the remainder when divide by 4 or 5 is 1, then the remainder when divided by 4*5=20 will also be 1.

Since 3,4 and 5 are relative primes we are looking for a number that has a 1 remainder when divided by 3*4*5 = 60. That number is 60+1 =61. 60 is divisible by 3 and 4 and 5, so 61 will give us 1 remainder.

There were 61 camels in the caravan.