Question

Four positive integers a, b, c, and d have a product of 8! and satisfy: ab + a + b = 524 bc + b + c = 146 cd + c + d = 104 What is a - d? (A) 4 (B) 6 (C) 8 (D) 10 (E) 12​
Answer with proper explanation

Answer 1

we can rewrite the three equations as follows:

(a+1)(b+1) & = 525

(b+1)(c+1) & = 147

(c+1)(d+1) & = 105 Note that

(a + 1)(b + 1)

= ab + a + b + 1

= 524 + 1

= 525

= and

(b + 1)(c + 1)

= bc + b + c + 1

= 146 + 1

= 147 = 

Since (a + 1)(b + 1) is a multiple of 25 and (b + 1)(c + 1) is not a multiple of

5,

it follows that a + 1 must be a multiple of 25.

Since a + 1 divides 525, a is

one of 24, 74, 174, or 524.

Among these only 24 is a divisor of 8!,

so a = 24.

This implies that b + 1 = 21, and b = 20.

From this it follows that c + 1 = 7

and c = 6. Finally,

(c + 1)(d + 1) = 105 = 3 · 5 · 7,

so d + 1 = 15 and d = 14.

Therefore, a − d = 24 − 14 = 10.

Answer 2

Given:

Four positive integers a, b, c, and d have a product of 8! and satisfy:

ab + a + b = 524, bc + b + c = 146, cd + c + d = 104

To Find:

What is a - d?

Solution:

We can rewrite the three equations as follows:

(a+1)(b+1)  = 525

(b+1)(c+1)  = 147

(c+1)(d+1)  = 105

Note that

(a + 1)(b + 1) = ab + a + b + 1 = 524 + 1 = 525

and

(b + 1)(c + 1) = bc + b + c + 1 = 146 + 1 = 147

Since (a + 1)(b + 1) is a multiple of 25 and (b + 1)(c + 1) is not a multiple of 5, it follows that (a + 1) must be a multiple of 25.

Since a + 1 divides 525, a is one of 24, 74, 174, or 524.

Among these only 24 is a divisor of 8!

So, a = 24.

This implies that b + 1 = 21, and b = 20.

From this it follows that c + 1 = 7

and c = 6.

Finally,

(c + 1)(d + 1) = 105 = 3 x 5 x 7,

So,

d + 1 = 15 and d = 14.

⇒a − d = 24 − 14 = 10

Hence, a - d is equal to 10 (option D).