Question

HELP ASAP PLEASE
A palindrome, such as 22022, is a number that remains the same when its digits are reversed. 22 is a two-digit palindrome. x is a three-digit palindrome. 22+x is a four-digit palindrome. What is the sum of the digits of x?

Answer 1

Solution:-

Given that

x is a three digit palindrome .

x must be 101,111,...,969,979,989,999

Given that 22+x is a four digit palindrome .

That means if we add x then we get four digit palindrome.

If x = 101 then x+22 = 101+22 = 123

.

.

.

If x = 969 then x+22 = 969+22 = 991

If x = 979 then x+22 = 979+22 = 1001

If x = 989 then x+22 = 989+22 = 1011

If x = 999 then x +22 = 999+22 = 1021

among these 1001 is the first four digit palindrome

So x must be equal to 979

x = 979 , a three digit palindrome

x+22 = 1001 ,a four digit palindrome

The sum of the digits in 979

= 9+7+9

= 25

Answer :-

The sum of the digits of x is 25

Answer 2

Sum of digit of x = 25 (9 + 7 + 9 ) if x is a 3 digit palindrome and 22 + x is a 4 digit palindrome

Given:

• x is a three Digit palindrome
• 22 + x is a four digit palindrome

To Find:

• Sum of the digits of x

Solution:

Sum of a 3 digit Number and 2 Digit number is 4 Digit number

Hence 3 digit Number must Start with 9

As 3 digit Number here is Palindrome hence number must end with 9

As x + 22 = 4 Digit number

Hence     978 ≤ x  ≤ 999  ( as 4 Digit number starts with 1000 )

Now 979 , 989 and 999 are 3 possible numbers

979 + 22 = 1001  

1001 is Palindrome Hence Satisfied

x = 979

989 + 22 = 1011  not a palindrome

999 + 22 = 1021  not a palindrome

Hence , x = 979

Sum of digit of x = 9 + 7 + 9 = 25