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PLAYING WITH NUMBERS

Numbers in General Form: Reversing the Digits of a 2-digit Number ab and then
• Generally, a 2-digit number ab can be Subtracting or adding:
written as 10a + b, where a and b are whole • Difference in each case is always a multiple of 9 and when
numbers and a ≠ 0. it is divided by 9, the remainder will be ‘0’ and the quotient
Example: 35 = 3 × 10 + 5 will always be the difference of the digits i.e., a – b or b – a.
• Generally, a 3-digit number abc can be Example: 35 and 53
written as 100a + 10b + c, where a, b, and 53 – 35 = 18; 18 ÷ 9 ⇒ R = 0 and Q = 5 – 3 = 2.
c are whole numbers and a  0. • The sum is always a multiple of 11 and when we divide it
Example: 215 = 2 × 100 + 1 × 10 + 5 by 11, the remainder will be ‘0’ and the quotient will be
the sum of the digits i.e., (a + b).
Example: 46 and 64
Letters for Digits/Cryptarithmetic 46 + 64 = 110; 110 ÷ 11 ⇒ R = 0 and Q = 6 + 4 = 10.
3 2 A 3 2 5 Reversing the Extreme Digits of a 3-digit Number abc and
2 A 1 2 5 1
+ 5 7 6 + 5 7 6 then Subtracting: The difference of any 3-digit number abc
and the number cba obtained by reversing its extreme digits,
Number Puzzle and Games: is completely divisible by 99 and the quotient is the difference
• Coding and Decoding of its extreme digits a and c (a ≠ c).
B R O T H E R S I S T E R Example: 127;
and
5 7 0 8 3 6 7 9 1 9 8 6 7 721 – 127 = 594 ÷ 99 ⇒ R = 0 and Q = 7 – 1 = 6.
HOBSTERS is 30598679 The sum of all three-digit numbers formed by reversing its
• Number patterns digits is divisible by 2, 3, 37, 111, 222 and sum of its digits.
2, 3, 5, 7, 11, 13, 17, 19, ... Example: 235; The numbers formed by interchanging the digits
of 235 are: 523, 325, 253, 532, 352
Pascal’s triangle Magic square Sum of these numbers = 235 + 523 + 325 + 253 + 532 + 352

1 18 11 16 45 = 2220 = 2 × 1110
1 1 13 15 17 45 = 2 × 111 × 10
1 2 1 14 19 12 45 = 3 × 740
1 3 3 1 = 222 × 10
1 4 6 4 1 45 45 45 45 45
= 37 × 60.

Divisibility Rules

A number is divisible by:
• 2 if its ones digit is an even number. Example: 100, 502, 404, 706, 1008, etc. are divisible by 2.
• 10 if its ones digit is 0. Example: 30, 6710, 1100, etc. are divisible by 10.
• 5 if the ones digit of a number is either 5 or 0. Example: 30, 6715, 1105, etc. are divisible by 5.
• 3 and 9 if the sum of digits is divisible by 3 and 9. Example: 3441 is divisible by 3, as 3 + 4 + 4 + 1 = 12, divisible
by 3. 5121 is divisible by both 9 and 3, as 5 + 1 + 2 + 1 = 9 divisible by both 9 and 3.
• 11 if the difference between the sum of its digits in odd places and the sum of its digits in even places is either
0 or a multiple of 11. Example: 605 is divisible by 11, as 6 + 5 – 0 = 11, divisible by 11.

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