Answer:
Explanation:
We need to find the number of teenagers in the group, given that the product of their ages is 18,727,200.
To solve this problem, we need to factorize the given number into its prime factors and then determine how many distinct factors there are.
18,727,200 can be factorized as:
18,727,200 = 2^6 × 3^2 × 5^2 × 13^2
To find the number of distinct factors, we add 1 to each exponent and then multiply them together:
(6+1) × (2+1) × (2+1) × (2+1) = 7 × 3 × 3 × 3 = 189
Therefore, there are 189 factors of 18,727,200, which means that there are 189 ways to multiply whole numbers together to get this number.
Since we want to find the number of teenagers in the group, we need to look for combinations of factors that result in whole numbers for the ages. We can start by dividing the total number of factors by 2 (since we are looking for pairs of factors) and then slowly increase the divisor until we find the smallest number that results in a whole number.
189 ÷ 2 = 94.5 (not a whole number)
189 ÷ 3 = 63 (not a whole number)
189 ÷ 4 = 47.25 (not a whole number)
189 ÷ 5 = 37.8 (not a whole number)
189 ÷ 6 = 31.5 (not a whole number)
189 ÷ 7 = 27 (a whole number)
Therefore, there are 27 pairs of factors that result in whole numbers for the ages. Each pair corresponds to a group of teenagers, and since each group has the same number of teenagers, there are 27 teenagers in the group.