A three digit number is equal to 17 times the sum of its digits; If th

1.	A three digit number is equal to 17 times the sum of its digits; If the digits are reversed, the new number is 198 more than the old number ; also the sum of extreme digits is less than the middle digit by unity. Find the original number.
Answer» $\huge\underline\frak{\fbox{AnSwEr :-}}$ Let the digit in hundreds place be x and that in unit place be y. ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ H ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀x T ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀x + y + 1 unit⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀y ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ $\implies$ the three digit number is 100x + 10(x + y + 1) + y = 100x + 10x + 10y + 10 + y = 110x + 11Y + 10 the SUM of the digits in the given number = x + (x + y + 1) + y = 2x + 2y + 1 From first condition Given number = 17 ´ (sum of the digits) $\implies$ 110x + 11y + 10 = 17 ´ (2x + 2y + 1) $\implies$ 110x + 11y + 10 = 34x + 34y + 17 $\implies$ 76x - 23y = 7 . . . (I) The number obtained by reversing the digits = 100Y + 10(x + y + 1) + x = 110y + 11x + 10 Given number = 110x + 11y + 10 From 2nd condition, Given number + 198 = new number. 110x + 11y + 10 + 198 = 110y + 11x + 10 99x - 99y = -198 $\implies$ x - y = -2 $\implies$ x = y - 2 . . . (II) Substitute this value of x in equation (I). $\implies$ 76(y - 2) - 23y = 7 $\implies$ 76y - 152 - 23y = 7 $\implies$ 53y = 159 $\implies$ y = 3 $\implies$ the digit in units place is = 3 Substitute this value in equation (II) $\implies$ x = y - 2 $\implies$ x = 3 - 2 = 1 $\implies$ x = 1 $\implies$ The digit in hundred’s place is 1 the digit in ten’s place is 3 + 1 + 1 = 5 $\implies$ the number is 153.

A three digit number is equal to 17 times the sum of its digits; If the digits are reversed, the new number is 198 more than the old number ; also the sum of extreme digits is less than the middle digit by unity. Find the original number.

Answer»

$\huge\underline\frak{\fbox{AnSwEr :-}}$

Let the digit in hundreds place be x and that in unit place be y.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

H ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀x

T ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀x + y + 1

unit⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀y

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\implies$ the three digit number is 100x + 10(x + y + 1) + y

= 100x + 10x + 10y + 10 + y = 110x + 11Y + 10

the SUM of the digits in the given number = x + (x + y + 1) + y = 2x + 2y + 1

From first condition

Given number = 17 ´ (sum of the digits)

$\implies$ 110x + 11y + 10 = 17 ´ (2x + 2y + 1)

$\implies$ 110x + 11y + 10 = 34x + 34y + 17

$\implies$ 76x - 23y = 7 . . . (I)

The number obtained by reversing the digits

= 100Y + 10(x + y + 1) + x = 110y + 11x + 10

Given number = 110x + 11y + 10

From 2nd condition, Given number + 198 = new number.

110x + 11y + 10 + 198 = 110y + 11x + 10

99x - 99y = -198

$\implies$ x - y = -2

$\implies$ x = y - 2 . . . (II)

Substitute this value of x in equation (I).

$\implies$ 76(y - 2) - 23y = 7

$\implies$ 76y - 152 - 23y = 7

$\implies$ 53y = 159

$\implies$ y = 3

$\implies$ the digit in units place is = 3

Substitute this value in equation (II)

$\implies$ x = y - 2

$\implies$ x = 3 - 2 = 1

$\implies$ x = 1

$\implies$ The digit in hundred’s place is 1

the digit in ten’s place is 3 + 1 + 1 = 5

$\implies$ the number is 153.

A three digit number is equal to 17 times the sum of its digits; If the digits are reversed, the new number is 198 more than the old number ; also the sum of extreme digits is less than the middle digit by unity. Find the original number.

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A three digit number is equal to 17 times the sum of its digits; If the digits are reversed, the new number is 198 more than the old number ; also the sum of extreme digits is less than the middle digit by unity. Find the original number.​

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A three digit number is equal to 17 times the sum of its digits; If the digits are reversed, the new number is 198 more than the old number ; also the sum of extreme digits is less than the middle digit by unity. Find the original number.