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# Write cubes of natural numbers up to $19$ and verify cubes of odd natural numbers are odd.

Hint:Natural numbers are all numbers $1,2,3,4$ etc. They are the numbers you usually count and they will continue on up to infinity. The result of multiplying a whole number by itself triple times, that is we can write in the form $x \times x \times x = {x^3}$, Also we can say that ${x^3}$ is the cube of $x$. Any numbers that cannot be exactly divided by $2$ whose last digit ends with $1,3,5,7,9$ are all odd numbers.Finally we get the required cubes of natural numbers and also verify the cubes of odd numbers are odd.

First we have to write Cube of natural numbers up to $19$ is evaluated as follows:
The result of multiplying a natural number by itself triple times, this is known as cube of the natural number.
We get,
${1^3} = 1 \times 1 \times 1 = 1$
${2^3} = 2 \times 2 \times 2 = 8$
${3^3} = 3 \times 3 \times 3 = 27$
${4^3} = 4 \times 4 \times 4 = 64$
${5^3} = 5 \times 5 \times 5 = 125$
${6^3} = 6 \times 6 \times 6 = 216$
${7^3} = 7 \times 7 \times 7 = 343$
${8^3} = 8 \times 8 \times 8 = 512$
${9^3} = 9 \times 9 \times 9 = 729$
${10^3} = 10 \times 10 \times 10 = 1000$
${11^3} = 11 \times 11 \times 11 = 1331$
${12^3} = 12 \times 12 \times 12 = 1728$
${13^3} = 13 \times 13 \times 13 = 2197$
${14^3} = 14 \times 14 \times 14 = 2744$
${15^3} = 15 \times 15 \times 15 = 3375$
${16^3} = 16 \times 16 \times 16 = 4096$
${17^3} = 17 \times 17 \times 17 = 4913$
${18^3} = 18 \times 18 \times 18 = 5832$
${19^3} = 19 \times 19 \times 19 = 6859$
Also we can obtain cubes of odd numbers that are odd.
Then we write the odd numbers of the natural numbers.
They are any number that cannot be divisible by $\;2$ and also the last digit ends with $1,3,5,7,9$ are all odd numbers.
So, we can verify cubes of odd natural numbers are odd.
${1^3} = 1 \times 1 \times 1 = 1$
${3^3} = 3 \times 3 \times 3 = 27$
${5^3} = 5 \times 5 \times 5 = 125$
${7^3} = 7 \times 7 \times 7 = 343$
${9^3} = 9 \times 9 \times 9 = 729$
${11^3} = 11 \times 11 \times 11 = 1331$
${13^3} = 13 \times 13 \times 13 = 2197$
${15^3} = 15 \times 15 \times 15 = 3375$
${17^3} = 17 \times 17 \times 17 = 4913$
${19^3} = 19 \times 19 \times 19 = 6859$

As it is very clear that the cube of odd natural numbers is odd.
Hence, we get the required answer and also verified.

Note:Upon taking a cube of a number the whole number is used $3$ times, just like the sides of a cube. Odd numbers are not divisible by $2$. This can also be adopted to verify whether the numbers are odd or even.