if four positive integers a, b, c, d have a product of 8!, and satisfy..... Please answer the question with necessary steps
Answers
The value of a - d is 10
Correct 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 then the value of a - d is (A) 4 (B) 6 (C) 8 (D) 10 (E) 12
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 :
The value of a - d is
(A) 4
(B) 6
(C) 8
(D) 10
(E) 12
Solution :
Step 1 of 2 :
Find the value of a , b , c , d
Here it is given that four positive integers a, b, c, and d have a product of 8! and satisfy:
ab + a + b = 524 - - - - - - (1)
bc + b + c = 146 - - - - - - (2)
cd + c + d = 104 - - - - - - (3)
From Equation 1 we get
ab + a + b + 1 = 524 + 1
⇒ (a + 1)(b + 1) = 525 = 3 × 5² × 7 - - - - (4)
From Equation 2 we get
bc + b + c + 1 = 146 + 1
⇒ (b + 1)(c + 1) = 147 = 3 × 7² - - - - - - (5)
From Equation 4 we get
(a + 1)(b + 1) is divisible by 25
From Equation 5 we get
(b + 1)(c + 1) is not divisible by 25
∴ (a + 1) is divisible by 25
∴ a + 1 will be one of 25 , 75 , 175 , 525
Consequently , a will be one of 24 , 74 , 174 , 524
But none of 74 , 174 , 524 divide 8!
∴ a can not be one of 74 , 174 , 524
∴ a = 24
From Equation 4 we get
(24 + 1)(b + 1) = 525
⇒ 25(b + 1) = 525
⇒ b + 1 = 21
⇒ b = 20
From Equation 5 we get
(b + 1)(c + 1) = 147
⇒ (20 + 1)(c + 1) = 147
⇒ 21(c + 1) = 147
⇒ c + 1 = 7
⇒ c = 6
From Equation 3 we get
cd + c + d = 104
⇒ 6d + 6 + d = 104
⇒ 7d + 6 = 104
⇒ 7d = 98
⇒ d = 14
Step 2 of 2 :
Find the value of a - d
From above we get
a = 24 , b = 20 , c = 6 , d = 14
∴ a - d = 24 - 14 = 10
So the value of a - d is 10
Hence the correct option is (D) 10
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