Question

a high school class with 30 seniors, 18 juniors and 12 sophomores is divided into groups for a field expedition. each group has the same number of juniors, seniors and sophomores. what is the greatest number of groups that can be formed? how many of each student is in a group?
2
two even numbers have the gcf of 150. find the smallest such numbers so that neither is a factor of the other.

Explanation:

First Problem (Grouping Students)

Step1: Find GCF of 30, 18, 12

Prime factorize:
\( 30 = 2 \times 3 \times 5 \)
\( 18 = 2 \times 3^2 \)
\( 12 = 2^2 \times 3 \)
GCF is \( 2 \times 3 = 6 \).

Step2: Determine groups and per-group counts

Number of groups = 6.
Seniors per group: \( \frac{30}{6} = 5 \)
Juniors per group: \( \frac{18}{6} = 3 \)
Sophomores per group: \( \frac{12}{6} = 2 \)

Second Problem (Even Numbers with GCF 150)

Step1: Express 150 as \( 2 \times 3 \times 5^2 \)

We need two even numbers (so include factor 2), GCF 150, and neither a factor of the other.

Step2: Choose coprime multipliers >1

Let multipliers be 2 and 3 (coprime, neither divides the other).
First number: \( 150 \times 2 = 300 \)
Second number: \( 150 \times 3 = 450 \)

Answer:

s:

1. Greatest number of groups: 6.

Seniors per group: 5, Juniors per group: 3, Sophomores per group: 2.

1. Smallest such numbers: 300 and 450.