Question

The square of which of the following would be an odd number?
A) 512
B) 320
C) 431
D) 220

Answer 1

Hint: We need to know the unit digit or ones place value of the number to identify them as odd numbers or even numbers.
Unit’s places of odd numbers are: 1, 3, 5, 7, 9.
Number 1567829, unit place digit is 9. Thus, this number is odd.
Unit’s place of even numbers are: 0, 2, 4, 6, 8.
Number 7894, unit place digit is 4. Thus, this number is even.
Square of a number means the product of numbers twice to itself. For example:
Let the number a, square of a is equal to
\[\mathop {\left( a \right)}\nolimits^2 = a \times a\]
One’s place or unit place is the leftmost digit of a number.
The power cycle of a number: the numbers have cyclicity (repetition) of their units’ digits for increasing powers.
For example:
$
  \mathop 4\nolimits^1 {\text{ }} = {\text{ 4}},{\text{ unit digit = 4}} \\
  \mathop 4\nolimits^2 {\text{ }} = {\text{ 16}},{\text{ unit digit = 6}} \\
  \mathop 4\nolimits^3 {\text{ }} = {\text{ 64}},{\text{ unit digit = 4}} \\
  \mathop 4\nolimits^4 {\text{ }} = 256,{\text{ unit digit = 6}} \\
 $
The power cycle of 4 = (4, 6)
The power cycle of 4 contains two numbers. Therefore, its cyclicity is 2.
Example question on use of cyclicity: find the unit place of $\mathop 4\nolimits^{251} $ .
Sol: The power cycle of 4 = (4, 6)
cyclicity of 4 = 2,
On dividing the power of 4 by cyclicity
251 divided by 2, remainder is 1.
$\mathop 4\nolimits^{251} {\text{ reduced to }}\mathop 4\nolimits^1 $
$\mathop 4\nolimits^1 = 4$, unit place = 4

Complete step-by-step answer:
Step 1: Calculate unit digit of square of 512
square of 512 $
  \mathop { = {\text{ 512}}}\nolimits^2 \\
   = 512 \times 512 \\
 $
Unit place of 512 = 2
The power cycle of 2:
$
  \mathop 2\nolimits^1 {\text{ }} = {\text{ }}2,{\text{ unit digit = 2}} \\
  \mathop 2\nolimits^2 {\text{ }} = {\text{ }}4,{\text{ unit digit = 4}} \\
  \mathop 2\nolimits^3 {\text{ }} = {\text{ }}8,{\text{ unit digit = 8}} \\
  \mathop 2\nolimits^4 {\text{ }} = 16,{\text{ unit digit = 6}} \\
  \mathop 2\nolimits^5 {\text{ }} = 32,{\text{ unit digit = 2}} \\
 $
Therefore, the power cycle of 2 = (2, 4, 6, 8)
The power cycle of 2 contains four numbers. Therefore, its cyclicity is 4.
Unit place of $\mathop {512}\nolimits^2 = $ unit place of $\mathop 2\nolimits^2 $
Dividing the power of 512 by cyclicity of 2
2 is divided by 4, remainder = 2 ($\because $ cyclicity of 2 is 4)
$\because \mathop 2\nolimits^2 $
Unit place of $\mathop 2\nolimits^2 = 4$
Unit place of $\mathop {512}\nolimits^2 $= unit place of \[\mathop 2\nolimits^2 = 4\]
$ \Rightarrow $ Unit place of \[\mathop {512}\nolimits^2 = 4\]
Thus, 512 is not an odd number.
Step 2: Calculate unit digit of square of 320
square of 320 $
  \mathop { = {\text{ 320}}}\nolimits^2 \\
   = 320 \times 320 \\
 $
Unit place of 320 = 0
The power cycle of 0:
\[
  \mathop 0\nolimits^1 {\text{ }} = {\text{ 0}},{\text{ unit digit = 0 }} \\
  \mathop 0\nolimits^2 {\text{ }} = {\text{ 0}},{\text{ unit digit = 0}} \\
  \mathop 0\nolimits^3 {\text{ }} = {\text{ 0}},{\text{ unit digit = 0}} \\
 \]
The power cycle of 0 = (0)
The power cycle of 0 contains only one number, i.e. 0. Therefore, no matter what is the power of 0, the unit digit will always be 0.
Unit place of $\mathop {320}\nolimits^2 $= unit place of \[\mathop 0\nolimits^2 = 0\]
$ \Rightarrow $ Unit place of \[\mathop {320}\nolimits^2 = 0\]
Thus, 320 is not an odd number.
Step 3: Calculate unit digit of square of 431
square of 431 $
  \mathop { = {\text{ 431}}}\nolimits^2 \\
   = 431 \times 431 \\
 $
Unit place of 431 = 1
The power cycle of 1:
\[
  \mathop 1\nolimits^1 {\text{ }} = {\text{ 1}},{\text{ unit digit = 1 }} \\
  \mathop 1\nolimits^2 {\text{ }} = {\text{ 1}},{\text{ unit digit = 1}} \\
  \mathop 1\nolimits^3 {\text{ }} = {\text{ 1}},{\text{ unit digit = 1}} \\
 \]
The power cycle of 1 = (1)
The power cycle of 1 contains only one number, i.e. 1. Therefore, no matter what is the power of 1, the unit digit will always be 1.
Unit place of $\mathop {431}\nolimits^2 $= unit place of \[\mathop 1\nolimits^2 = 1\]
$ \Rightarrow $ Unit place of \[\mathop {431}\nolimits^2 = 1\]
Thus, 431 is an odd number
Step 4: Calculate unit digit of square of 220
square of 220 $
  \mathop { = {\text{ 220}}}\nolimits^2 \\
   = 220 \times 220 \\
 $
Unit place of 220 = 0
The power cycle of 0 = (0) (from step 2)
The power cycle of 0 contains only one number, i.e. 0. Therefore, no matter what is the power of 0, the unit digit will always be 0.
Unit place of $\mathop {220}\nolimits^2 $= unit place of \[\mathop 0\nolimits^2 = 0\]
$ \Rightarrow $ Unit place of \[\mathop {220}\nolimits^2 = 0\]
Thus, 220 is not an odd number.

Therefore, the square of 431 is an odd number. Thus, the correct option is (C).

Additional information: In the Indian numeral system, the place values of digits go in the sequence of Ones, Tens, Hundreds, Thousands, Ten Thousand, Lakhs, Ten Lakhs, Crores and so on.
For example: write the place value of all the digits of the number: 2,45,34,720.

Place value	Digit
Ones	0
Tens	2
Hundreds	7
Thousands	4
Ten-thousands	3
Lakhs	5
Ten-lakhs	4
Crore	2

Note:
The power cycle and cyclicity, to find the unit place of number, are more useful for numbers with large powers.
 Students can learn the power cycle of other numbers as well. Example given:
Power cycle of 3:
$
  \mathop 3\nolimits^1 {\text{ }} = {\text{ 3}},{\text{ unit digit = 3}} \\
  \mathop 3\nolimits^2 {\text{ }} = {\text{ 9}},{\text{ unit digit = 9}} \\
  \mathop 3\nolimits^3 {\text{ }} = {\text{ 27}},{\text{ unit digit = 7}} \\
  \mathop 3\nolimits^4 {\text{ }} = {\text{ 81}},{\text{ unit digit = 1}} \\
  \mathop 3\nolimits^5 {\text{ }} = 243,{\text{ unit digit = 3}} \\
 $
The power cycle of 3 = (3, 9, 7, 1)
The power cycle of 3 contains four numbers. Therefore, its cyclicity is 4.
Power cycle of 5:
$
  \mathop 5\nolimits^1 {\text{ }} = {\text{ 5}},{\text{ unit digit = 5}} \\
  \mathop 5\nolimits^2 {\text{ }} = {\text{ 25}},{\text{ unit digit = 5}} \\
  \mathop 5\nolimits^3 {\text{ }} = {\text{ 125}},{\text{ unit digit = 5}} \\
  \mathop 5\nolimits^4 {\text{ }} = 625,{\text{ unit digit = 5}} \\
 $
The power cycle of 5 = ( 5 )
The power cycle of 5 contains only one number, i.e. 5. Therefore, no matter what is the power of 5, the unit digit will always be 5.
Odd numbers can be expressed in the form of $\left( {2n - 1} \right)$, where $n$is a natural number.
Example given: $431 = \left[ {2(216) - 1} \right]$
Even numbers can be expressed in the form of $\left( {2n} \right)$, where $n$is a natural number.
Example given: $320 = 2\left( {160} \right)$.