if a collection of pencils is placed in rows of 3, there are 2 pencils left; if placed in rows of 4, there are 2 left; and if placed in rows of 7, there are 2 left. what is the smallest possible number of pencils in the collection

Answer 1

To find the smallest possible number of pencils in the collection, we need to find the least common multiple (LCM) of the three numbers: 3, 4, and 7.

The prime factorization of 3 is 3.
The prime factorization of 4 is 2^2.
The prime factorization of 7 is 7.

The LCM is the product of the highest powers of all prime factors involved. Therefore, LCM(3, 4, 7) = 2^2 × 3 × 7 = 84.

So, the smallest possible number of pencils in the collection is 84.