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## Saturday, May 30, 2020

### ODD AND EVEN NUMBERS - FUN WITH THEM , ACTIVITY BASED AND CHALLENGING QUESTIONS

ODD AND EVEN NUMBERS

In order to understand about odd and even numbers, lets conduct an activity. Below are some counters, and they are to be paired

So 42 is an even number. Even numbers are those numbers which can be paired, without leaving any number alone. Suppose the number is 10, so 2 + 2 + 2 + 2 + 2 = 10, there are 5 pairs of two each, and any number is not left alone. Therefore 10 is also an even number .

Hence 39 is an odd number. Odd numbers are those numbers which cannot be exactly divided into groups of two each or paired. The number 15 is also an odd number as 2 + 2 + 2 + 2 + 2 + 2 + 2 + 1 = 15, there are 7 even pairs of two each plus one is 15.

So it is clear that even numbers are those numbers, which are divisible by 2 exactly, and odd numbers are those are not divisible by 2 exactly. As 42 when divided by 2 or paired, it is 21, with no remainder, so divisible by 2 exactly. Therefore 42 is an even number. But when we divide 39 by 2 it is 19, with remainder 1, not divisible by 2 exactly. So 39 is an odd number.

Even numbers are those numbers whose unit digit is 2, 4, 6, 8 or 0. This way we can recognize just by observing its unit digit, whether it is a small or huge number. Suppose 46 is an even number, the answer is 23 with no remainder. Considering 4,05,768 and verifying it 4,50,768/2 = 225383 with no remainder it is an even number.

Odd numbers are those unit digit is 1, 3, 5, 7 and 9. It does not matter that how much bigger the number may be, if its unit digit is one of 1, 3, 5, 7, 9 it is an odd number. For instance 59 is an odd number, as unit digit is 9. Verifying it when we divide 59 by 2 or 59/2 the answer is 29 and remainder is 1. The number as larger as 2,30,905 is also an odd number, because its unit digit, or the last digit is 5. Dividing 2,30,905 by 2 the answer is 1,15,452 and remainder is 1. Thus not divisible by 2 exactly, is an odd number.

Therefore one method of differentiating between odd and even number is seeing its unit digit, that is last digit from the extreme right side of the number. This is a quick method. Another method is checking divisibility by 2, which takes some time.

INTERESTING PATTERNS IN ODD AND EVEN NUMBERS WHILE ADDING

When two odd numbers are added, the result is an even number. That is when odd numbers are added even times, it is an even number always. Check this fact and add 23 + 23 + 23 + 23 + 23 + 23, that is 23 is added six times which is even times, the answer would be 138 an even number. If the different odd numbers are added even times, then also it is an even number as shown.

In even numbers, no matter how many times you add even number, it would always result in an even number finally. Cross verify it by adding 48 five times, that is 48 + 48 + 48 + 48 + 48 = 120.  Add 28 eight times or different even numbers any number of times it is an even number, try it.

Thus the game is played with odd numbers, not with even numbers.

ODD AND EVEN NUMBERS IN SUBTRACTION -

When an odd number is subtracted from an odd number it is an even number, as two odds is an even number.  If an even number is subtracted from an odd number or in reverse case, it is always an odd number.

If an even number is subtracted from an even number, the answer is an even number. No matter how many even numbers we add and then series of even numbers is subtracted from it, the result would never be an odd number, and always an even number. For instance the numbers 32, 44 and 26 are added it is 102, and then from it 24 and 38 is subtracted, that is 102 – 24 – 38 = 50.

So the manipulation here also is with odd numbers, not with even numbers.

If an even number is subtracted from addition of two odd numbers, it is an even number, as two odds is even. But if an even number is subtracted from addition of three odds, it is an odd number, as odd number is added odd number of times.

PATTERNS IN ODD AND EVEN NUMBER WHILE MULTIPLYING

(A) ONLY ODD NUMBERS MULTIPLICATION

Odd number   X    odd number       =  Odd number

11             X         11                     =     121

17            X        13                       =     221

Odd no X odd no X odd no X odd no X odd no = Odd no

9    X     9       X      9      X     9        X     9   = 59,049

7    X     3       X    15       X    5         X   21   = 33,075

1.The same odd number is multiplied odd number of times or even number of times, the result is always an odd number.

2. Different odd numbers are multiplied odd number or even number of times, in this case also answer is an odd number always.

(B) ONLY EVEN NUMBERS MULTIPLICATION

Even number   X    even number     =  Even number

12         X       12                             =    144

8         X       14                              =    112

Even no X even no X even no X even no X even no = Even no

6   X    6    X   6     X     6     X   6     = 7,776

2   X    8    X   16    X   10    X   18   = 46,080

1.The same even number is multiplied odd number of times or even number of times, every time the result is an even number.

2. Different even numbers are multiplied odd number or even number of times,in this case also the solution is an even number always.

(C) ODD AND EVEN NUMBERS MULTIPLICATION

Even number   X    odd number      =  Even number

24              X       15                   =     360

Even number X even number X odd number = Even number

8           X          12          X       13             = 1,248

Even number X odd number X odd number = Even number

16            X           7            X        9             = 1,008

Even no X  odd no X   odd no X  odd no        = Even number

22      X     7       X         11   X     9              = 15,246

When an odd number is multiplied with an even number, it is always an even number. Multiplication is repeated addition, so alternatively suppose 27 an odd number, is added 6 number of times, or 27 X 6 = 162 is an even number.

If odd number one time or more than one time, is multiplied by an even number the result is always an even number. Whether the odd number is multiplied one time, two times, six times or how many times by an even number, in every case it is an even number. It is also an even number when different odd numbers are multiplied by an even number.

QUESTIONS BASED ON ODD AND EVEN NUMBERS

## Q – 1.Which of the following option would be an odd number ?(a) 123 X 12    (b) 23 + 46 + 34(c) 8 X 24  (d) 54 + 64 + 57 + 47Q - 2. Ram receives one note of odd number daily from his father.The total amount that he would collect in the month of April is an  (a) Even number  (b) Odd number (c) Cannot be determined  (d) None of these

Q - 3. Shekhar plays cricket on every alternate day in the month of November starting from 4th for the whole month. How many times he would play cricket in that month, and it is an even or odd number ?

(a) 16 times, even number

(b) 15 times, odd number

(c) 14 times, even number

(d) None of these

Q - 4.The number lies between 40 and 100 and is an even number. It is a multiple of 9 and 5. The number is

(a) 81

(b) 85

(c) 90

(d) 45

Q - 5. The sum of four consecutive numbers is -

(a) Always even

(b) Always odd

(c) Can be odd or even

(d) Cannot be determined

Q - 6.The difference between the sum of two even  numbers and sum of three odd numbers , assuming that the sum of two even numbers is greater than the sum of three odd numbers is

(a) Always even

(b) Always odd

(c) Can be odd or even

(d) None of these

Q – 7.Which of the following statement is not true about even numbers ?

(a) Even numbers end in 0, 2, 4, 6 and 8

(b) All the even numbers are exactly divisible by 2

(c) No matter how many times we add even numbers, the result
is always an even number

(d) All the even numbers are exactly divisible by 3

Q – 8.Which is correct regarding odd numbers ?

(a) If we multiply an odd number, even number of times, it is an
even number

(b) All the odd numbers are exactly divisible by 5

(c) If we add different odd numbers, even number of times, the
result is an odd number

(d) If we add different odd numbers, odd number of times,
ultimately an odd number is obtained

Q - 9.There is an odd-even system for the vehicles, on odd date
only odd number of vehicle runs. Raj's car is of odd number,
which he drives to his office. Sunday is his off  day. For how
many days would he drive his car to the office in May starting

from 1st May ?

(a) 14

(b) 12

(c) 16

(d) 15

Q -10. The multiples of number 13 can be -

(a) Only odd number

(b) Only even number

(c) Either odd or even number

(d) None of these

Q - 1. (b) 123 X 12 is an even number, as odd number is multiplied with an even number. 23 + 46 + 34 = 103 is an odd number, two even numbers are added in an odd number. 8 X 24 is an even number, as an even number is multiplied with an even number. 54 + 64 + 57 + 47 = 222 is an even number, two evens is even and two odds is also even, so even + even = even .

Q - 2. (a) Note Ram gets from his father is an odd number. There are even number of days that is 30 in the month of April. Amount he would collect in the month of April is Amount of note  is = odd X number of days = even. Odd X even = Even

Q – 3. (c). Shekhar starts to play cricket from 4th June, and plays on every alternate day, so he would play on 4th, 6th, 8th, 10th, 12th, 14th, 16th, 18th, 20th, 22nd, 24th, 26th, 28th and 30th June which is 14 days. 14 is an even number. June has 30 days.

Q - 4. (c) 81 is a multiple of 9, but not of 5. 85 is a multiple of 5, but not of 9. 90 is a multiple of 9 and 5 and lies between 40 and 100 and is an even number also. 45 is a multiple of 9 and 5, but is an odd number.

Q - 5. (a) The sum of four consecutive numbers is always even. Let us take four consecutive numbers 24, 25, 26 and 27. There are two odds that is 25 and 27 and two evens that is 24 and 26. Two odds is even two evens is even, so even + even = even. If we start from the odd number like 33, 34, 35 and 36 then also there are two odds and two evens.

Q - 6. (b) The sum of two even numbers suppose 48 and 72 is 120 which is even. The sum of three odd numbers, for instance 13, 27 and 33 is 73 which is odd. Even - odd is odd always or 120 - 73 = 43 is odd .

Q - 7. (d) Even numbers end in 0, 2, 4, 6 or 8 is true. All the even numbers are exactly divisible by 2 is also correct, as 246, 328 even numbers are exactly divisible by 2. How many times we add an even number is an even number is also correct. Suppose 38 is added five times it is 190 an even number. All the even numbers are not exactly divisible by 3. For instance 22, 46 is not divisible by 3 exactly .

Q - 8. (d) Whether an odd number is multiplied odd number of times, or even number of times the result is always an odd number, not an even number. For instance 3 is multiplied 4 number of times, it is 81 an odd number. All the odd numbers are not exactly divisible by 5. For example 49, 51 is not divisible by 5. If we add different odd numbers even number of times, the result is not an odd number, but an even number. For instance 35, 47, 53 and 61 added is 196 an even number. But when we add different odd numbers, odd number of times like 21, 19, 39, 71 and 81 added is 231 an odd number .

Q - 9.(a) It is 14 days. Starts from 1st which is odd and is a Tuesday, then 3rd Thursday, 5th Saturday, 7th Monday, 9th Wednesday, 11th Friday, (13th Sunday off day so not counted ), 15th Tuesday, 17th Thursday, 19th Saturday, 21st Monday, 23rd Wednesday, 25th Friday, ( 27th Sunday again off day ), 29th Tuesday, 31st Thursday. Total is 16 days and two Sundays excluded, so it is 14 days .

Q - 10. (c) The multiple of number 13 can be either odd or even number. For instance 13 X 3 = 39 is an odd number, whereas 13 X 8 = 104 is an even number .

(b