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I have posted a video on YouTube to discuss Even and Odd Numbers
Attached pdf of this Article as SPOILER at the top! Happy learning!
Following is Covered in the Video
Theory of Even and Odd Numbers
⁍ What are Even Numbers ? ⁍ Even Number Problem ? ⁍ What are Odd Numbers ? ⁍ Odd Numbers Problem ? ⁍ Properties of Even and Odd Numbers ? ⁍ Solved Problems
What are Even Numbers
A number which gives 0 as remainder when divided by 2 is an even number.
⁍ Example: 18
⁍ Even numbers end with a units’ digit of 0, 2, 4, 6, 8 ⁍ An even number “n” is represented as n = 2k , where k is an integer
⁍ Example: Consecutive even numbers can be taken as 2k-4, 2k-2, 2k, 2k+2, 2k+4
Even Numbers Problem: Sum of three consecutive even numbers is 60. Find the numbers?
Sol: Let the three numbers be 2k-3, 2k-1, 2k+1, 2k+3 => 2k-3 + 2k-1 + 2k+1 + 2k+3 = 80 => 8k = 80 => k = 10 => Numbers are 17, 19, 21, 23
Properties of Even and Odd Numbers
Addition and Subtraction
-Addition E + E = E E + O = O O + E = O O + O = E Adding Odd number of Odds will give us O Adding even number of Odds will give us E
where E -> Even, O -> Odd
-Subtraction E – E = E E – O = O O – E = O O – O = E Subtracting odd number of Odds will give us O Subtracting even number of Odds will give us E
where E -> Even, O -> Odd
Division and Multiplication
-Division E / E = E or F or O E / O = E or F O / E = F O / O = O or F
where E -> Even, O -> Odd, F -> Fraction
-Multiplication E * E = E => \(E^{+ve Integer}\) = E E * O = E O * E = E O * O = O => \(O^{+ve Integer}\) = O
where E -> Even, O -> Odd
⁍ Product of numbers will be even when there is at least one number is Even.
Example: 3*3*2 = 18 = Even as there was one even number 2 on the left side
⁍ Product of numbers will be odd ONLY when all numbers are odd.
Example: 3*3*3 = 27 = Odd as all the numbers on the left side were odd
Solved Problems
Q1. If x, y, and z are integers and x + yz is odd, then which of the following must be true? I. x + z is even II. x + y is odd III. y + z is odd IV. xy is even V. yz is even VI. xz is odd VII. xyz is even
Either x is E and yz is O. => Only one case possible Case 1: x = E, y = O, z = O
Or x is O and yz is E => Three cases possible Case 2: x = O, y = E, z = O Case 3: x = O, y = O, z = E Case 4: x = O, y = E, z = E
I. x + y is odd We have to check for x + y in all the 4 possible cases For x + y to be odd, one has to even and other has to be odd But in 3rd case both x and y are odd. So, not possible II. y + z is odd Similar logic, in case 1 and case 4 its not possible III. x + z is even Similar logic, in case 1, 3,4 its not possible IV. xy is even At least one has to even. Not possible in case 3 V. yz is even At least one has to even. Not possible in case 1 VI. xz is odd Both have to odd. Not possible in case 1,3,4 VII. xyz is even At least one has to even. Possible in all the cases
Answer VII
Q2.If x is even, y is odd, z is even, then whether the following are odd or even I. x + yz II. x + y + yz III. xy + z IV. (x+1)*(y+1)*(z+1) V. xy*(z+1) VI. \((x+1)^2*y*(z+1)^3\)
Sol: I. E + O*E = E + E = E II. E + O + O*E = E + O + E = O III. E*O + E = E + E = E IV. (E + O) * (O + O) * (E + O) = O ( E * O = E V. E *O * (E + O) = E VI. \((E+O)^2*O*(E+O)^3\) = O * O * O = O
Q3. Product of 4 consecutive numbers will be divisible by all of the following EXCEPT? A. 6 B. 8 C. 12 D. 24 E. 48?
Sol: Let's take values to solve this. Lets take 4 numbers as 1, 2, 3, 4 Their product = 1*2*3*4 = 24 and will be divisible by all numbers except E Answer E
Theory: Product of n consecutive numbers will always be divisible by n!
Q4. Sum of three consecutive even numbers is divisible by all of the following EXCEPT A. 1 B. 2 C. 3 D. 4 E. 6
05 Jan 2024, 21:14
