1. **State the problem:** Given two numbers 156 and 288, we need to find: (a) their greatest common factor (g.c.f.), (b) their least common multiple (l.c.m.), (c) the quotient of their product divided by the g.c.f., (d) a conclusion based on the above results. 2. **Find the g.c.f. (greatest common factor):** We use the prime factorization method. Prime factors of 156: $$156 = 2 \times 78 = 2 \times 2 \times 39 = 2^2 \times 3 \times 13$$ Prime factors of 288: $$288 = 2 \times 144 = 2 \times 2 \times 72 = 2^4 \times 3^2$$ The g.c.f. is the product of the lowest powers of common prime factors: $$\text{g.c.f.} = 2^{\min(2,4)} \times 3^{\min(1,2)} = 2^2 \times 3^1 = 4 \times 3 = 12$$ 3. **Find the l.c.m. (least common multiple):** The l.c.m. is the product of the highest powers of all prime factors: $$\text{l.c.m.} = 2^{\max(2,4)} \times 3^{\max(1,2)} \times 13^{\max(1,0)} = 2^4 \times 3^2 \times 13 = 16 \times 9 \times 13$$ Calculate: $$16 \times 9 = 144$$ $$144 \times 13 = 1872$$ 4. **Divide the product of the two numbers by the g.c.f.:** Calculate the product: $$156 \times 288 = 44928$$ Divide by g.c.f.: $$\frac{44928}{12} = \frac{\cancel{12} \times 3744}{\cancel{12}} = 3744$$ 5. **Formulate a conclusion:** Compare the quotient with the l.c.m.: $$3744 \text{ (product/g.c.f.)} \quad \text{and} \quad 1872 \text{ (l.c.m.)}$$ Since $3744 > 1872$, the product divided by the g.c.f. is greater than the l.c.m. **Final answers:** (a) g.c.f. = 12 (b) l.c.m. = 1872 (c) product/g.c.f. = 3744 (d) The product of the two numbers divided by the g.c.f. is greater than the l.c.m.