We are given that $X = a^2 \times 7^{b+2}$ and $Y = a^3 \times 7^2$. We are also given that the greatest common factor (GCF) of $X$ and $Y$ is 1225 and the least common multiple (LCM) of $X$ and $Y$ is 42875. We need to find the values of $X$ and $Y$.

1. Problem Description

We are given that $X = a^2 \times 7^{b+2}$ and $Y = a^3 \times 7^2$.

We are also given that the greatest common factor (GCF) of $X$ and $Y$ is 1225 and the least common multiple (LCM) of $X$ and $Y$ is

4

2

8

7

5. We need to find the values of $X$ and $Y$.

2. Solution Steps

First, we express GCF and LCM in terms of prime factors.

We know that $1225 = 5^2 \times 7^2$ and $42875 = 5^3 \times 7^2 \times 5^0 \times 7^0$

$42875 = 5^3 \times 7^3$.

Since $GCF(X, Y) = a^{\min(2, 3)} \times 7^{\min(b+2, 2)} = a^2 \times 7^{\min(b+2, 2)} = 5^2 \times 7^2$, we have $a = 5$ and $\min(b+2, 2) = 2$.

Since $LCM(X, Y) = a^{\max(2, 3)} \times 7^{\max(b+2, 2)} = a^3 \times 7^{\max(b+2, 2)} = 5^3 \times 7^3$, we have $a=5$ and $\max(b+2, 2) = 3$.

Therefore, $a=5$.

$\min(b+2, 2) = 2$ means $b+2 \ge 2$, so $b \ge 0$.

$\max(b+2, 2) = 3$ means that either $b+2=3$ or $2=3$, but $2=3$ is incorrect, thus $b+2 = 3$, so $b = 1$.

Then, $X = a^2 \times 7^{b+2} = 5^2 \times 7^{1+2} = 5^2 \times 7^3 = 25 \times 343 = 8575$.

And $Y = a^3 \times 7^2 = 5^3 \times 7^2 = 125 \times 49 = 6125$.

3. Final Answer

$X = 8575$

$Y = 6125$