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Example 3: Sum of the digits of a 2-digit number is 8. If the digits are reversed, the new number
so formed decreases by 18. Find the number.

Solution: Let the ones digit of the number be x.
Then, the tens digit = 8 – x

Original number = 10 × (8 – x) + 1 × x

= 80 – 10x + x = 80 – 9x
On reversing the digits, the new number formed = 10 × x + 1 × (8 – x) = 9x + 8

According to the given condition, we have
New number = Original number – 18

⇒ 9x + 8 = 80 – 9x – 18
⇒ 18x = 54

⇒ x = 3
Thus, the required number is 80 – 9x = 80 – 9 × 3 = 53.

Letters for Digits/Cryptarithms

Sometimes, we come across puzzles in which we asked to find missing numbers represented by
alphabets in problems involving any of the four operations (+, –, ×, ÷). It is just like cracking a code.
A Cryptarithms is a type of mathematical puzzle in which the digits are replaced by letters of the
alphabet or other symbols. The solution involves finding the original digits. Making and solving
such puzzles are known as Cryptarithmetic or Cryptarithms. While solving Cryptarithms, some
conventions made are as follows:

(a) Each letter or symbol in a puzzle stands for just one digit and is represented by just one letter.
(b) The first digit of a number cannot be zero.

For example, sixty-five will be written as 65, not as 065 or 0065.

(c) When letters are replaced by their digits, the resultant arithmetical operation must be correct.
Example 4: Replace A, B, C, D by suitable numerals in the following addition:

(a) 3 2 A (b) 6 A B 5 (c) A
2 A 1 D 5 8 C + A
+ 5 7 6 + 9 3 5 1 + A
B A
Solution: (a) Here, the addition in the units column:

A + 1 = 6
Therefore, A = 5

So, the puzzle can be solved as below:

3 2 5
2 5 1
+ 5 7 6

341 Playing With Numbers