A three digit no. is equal to 17 times the some of its digits. If 198 is added to the no. the digits are interchanged. The addition of the 1st and the 3rd digit is 1 less than the middle digit. Find the no.
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Answers
Answer:
153
Step-by-step explanation:
Let it be abc.
Then, its numerical position will be 100a + 10b + c.
(i)
Given that three digit number is equal to 17 times the some of its digits.
⇒ 100a + 10b + c = 17(a + b + c)
⇒ 100a + 10b + c = 17a + 17b + 17c
⇒ 83a - 7b - 16c = 0
(ii)
Given that if 198 is added to the number, the digits are reversed.
⇒ 100a + 10b + c + 198 = 100c + 10b + a
⇒ 100a - a + 10b - 10b + c - 100c + 198 = 0
⇒ 99a - 99c+ 198 = 0
⇒ a - c + 2 = 0
⇒ c = a + 2
(iii)
Given that the addition of the 1st and the 3rd digit is 1 less than middle one.
⇒ a + c = b - 1
⇒ a - b + c + 1 = 0
Substitute (ii) in (i), we get
⇒ 83a - 7b - 16(a + 2) = 0
⇒ 83a - 7b - 16a - 32 = 0
⇒ 67a - 7b = 32 --------------------- (iv)
Substitute (ii) in (iii), we get
⇒ a - b + c + 1 = 0
⇒ a - b + a + 2 + 1 = 0
⇒ 2a - b + 3 = 0
⇒ b = 2a + 3 ----------------------- (v)
Substitute (v) in (iv), we get,
⇒ 67a - 7b = 32
⇒ 67a - 7(2a + 3) = 32
⇒ 67a - 14a - 21 = 32
⇒ 53a = 53
⇒ a = 1.
Substitute a = 1 in (v), we get
⇒ b = 2a + 3
= 5
Substitute b = 5 in (iii), we get
⇒ a - b + c + 1 = 0
⇒ 1 - 5 + c + 1 = 0
⇒ c = 3
Therefore, the number is 153.
Hope it helps!