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?
Show your thinking using pictures, words, or numbers.

The fact that when the camels walked in rows of 2 one walked alone means tha the number of camels divided by 2 leaves a remainder of 1. Also, that means that if you subtract 1 from the number of camels, the new number is a multiple of 2. This is:

The number of camels less 1 is a multiple of 2.

With the same reasoning, you obtain:

The number of camels less 1 is a multiple of 3.

The number of camels less 1 is a multiple of 4.

The number of camels less 1 is a multiple of 5.

Now, the numbers of camels less 1 is a multiple of 2, 3, 4, and 5. Now, let's search the first (least) common multiple of 2, 3, 4, and 5.

For that, you have to factor each number into its prime factors:

2 = 2
3 = 3
4 = 2²
5 = 5

The least common multiple is the product of the common and uncommon factors, each raised to its highest power: 2² × 3 × 5 = 60.

Remember that 60 is the number of camels less 1, thus the number of camels is 60 + 1 = 61.

The next common multiple is 2 × 60 = 120, which exceeds 100 thus the only possible value is the one determined above.

Now you can prove that 61 meets the conditions:

61 ÷ 2 = 30, plus one (remainder): means that 60 camels walked in 30 rows of 2 and one walked alone.

61 ÷ 3 = 20, plus one: 60 camels walked in 20 rows of 3 and one walked alone.

61 ÷ 4 = 15, plus one: 60 camels walked in 14 rows of 4 and one walked alone.

61 ÷ 5 = 12, plus one: 60 camels walked in 12 rows of 5 and one walked alone.