Question

Write cubes of natural numbers up to \[19\] and verify cubes of odd natural numbers are odd.

Answer 1

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.

Complete step-by-step answer:
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.