Playing with Numbers

The world of numbers is not only about calculations but also about exploration, patterns and clever puzzles that engage our minds in a delightful game. Playing with numbers invites us to think creatively, observe patterns and apply logic to solve puzzles that tickle our intellectual curiosity. In this chapter, let's delve into the exciting domain of playing with numbers and discovering the problems. Let's learn about the divisibility tests and rules that make it a captivating mathematical adventure.

General Form of Two-Digit Numbers

• A two-digit number is represented as “10a + b” where ‘a’ is the digit at the tens place and ‘b’ is the digit at the units place.
• If you reverse the digits, you get “10b + a”.
• Example: The two-digit number 27 is written as (10 × 2 + 7). By reversing the digit, we get 72 which is written as (10 × 7 + 2).

Interesting Facts about Numbers

Sum and Divisibility by 11:

• When you add a two-digit number to its reversed form, the result is always divisible by 11.
• Example:  The two-digit number is 27 and its reverse form is 72.
  ∴ Sum of the given number and its reversed form
  = 27 + 72
  = 99 (Which is divisible by 11)

Difference and Divisibility by 9:

• If you subtract the reversed two-digit number from the original, the result is always divisible by 9.
• Example:  The original number is 27 and its reverse form is 72.
  ∴ Subtraction of the reversed two-digit number from the original
  = 72 − 27
  = 45 (Which is divisible by 9)

Three-Digit Numbers

• A three-digit number (abc) is represented as "100a + 10b + c" where 'a', 'b' and 'c' are the digits at the hundreds, tens and units places, respectively.
• For a three-digit number (abc), the difference of a three-digit number from the number obtained by reversing the order of the digits is always divisible by ‘99’.
  Example: The three-digit number is 127 and its reverse form is 721.
  ∴ Difference of a three-digit number from its reverse form
  = 721 − 127
  = 594 (Which is completely divisible by 99)

Special Property for Three-Digit Numbers:

• The sum of three-digit numbers formed by different arrangements of the digits (abc, bca, cab) is always divisible by 37.
• Example: The three-digit number is 127.
  ∴ Sum of three-digit numbers formed by different arrangements of the digits = 127 + 271 + 712
  = 1110 (Which is completely divisible by 37)

Letters for Digits

We replace digits with letters in addition, subtraction, multiplication and division problems. The challenge is to figure out which letter corresponds to which digit like solving a code.

The two important rules to solve these problems are as follows:

1. One Letter represents One Digit:

Each letter used in the problem represents only one digit and each digit is represented by only one letter.

2. No Leading Zeros:

The first digit in a number cannot be zero. For example, if we have the number twenty-seven, we write it as 27 not as 027 or 0027.

Example: What is the sum of x, y and z in the given problem?

a) 13
b) 15
c) 21
d) 27

Answer: b) 15

Explanation: The addition is shown as:

Sum of x, y and z in the given problem = x + y + z
                                                          = 8 + 2 + 5
                                                           = 15

Divisibility Tests

Divisibility by 2: A number is divisible by 2 if the unit digit is 0, 2, 4, 6 or 8.

Divisibility by 3: A number is divisible by 3 if the sum of the digits is divisible by 3.

Divisibility by 4: A number is divisible by 4 if the last two digits are divisible by 4.

Divisibility by 5: A number is divisible by 5 if the unit digit is 0 or 5.

Divisibility by 6: A number is divisible by 6 if it is divisible by both 2 and 3.

Divisibility by 7: A number is divisible by 7 if the last digit of the number is doubled and subtracted from the remaining part, then their difference is 0 or divisible by 7.

Divisibility by 8: A number is divisible by 8 if the last three digits of the number are divisible by 8.

Divisibility by 9: A number is divisible by 9 if the sum of the digits is divisible by 9.

Divisibility by 10: A number is divisible by 10 if the unit digit is 0.

Divisibility by 11: A number is divisible by 11 if the difference between the sum of digits at odd places and even places should be either 0 or divisible by 11.

Divisibility by 12: A number is divisible by 12 if it is divisible by both 3 and 4.