Question: Divide 45 into three parts which are in AS such that their product is 1875.

Solution:

Let the three numbers which are in AS be (a-d), a and (a +d).

According to the question,
$or, a-d + a + a +d = 45$

$or, 3a = 45$

$or, a = \frac{45}{3}$

$\therefore a = 15$


Also,
$(a -d)(a)(a +d) = 1875$

$or, a (a² - d²) = 1875$

$or, 15(15^2 - d^2) = 1875$

$or, 225 - d^2 = \frac{1875}{15}$

$or, 225 - d^2 = 125$

$or, d^2 = 225 - 125$

$or, d^2 = 100$

$or, d = \sqrt{100}$

$\therefore d = \pm 10$


When d = +10,
$(a - d) = (15-10) = 5$
$ a = 15$
$(a + d) = (15+10) = 25$

When d = -10,
$(a -d) = (15+10) = 25$
$a = 15$
$or, (a + d) = (15-10) = 5$


Hence, the three parts in which 45 was divided are either 5,15,25 or 25,15,5.

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