Question

product of 4 positive integers a,b,c&d is 8!
If:
ab+a+b=524
bc+b+c=146
cd+c+d=104

find the value of a-d.(with steps)​

Answer 1

Answer:

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.

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