Answer :
To determine which of the given numbers are cubes of odd natural numbers, we need to understand the concept of a cube. A cube of a number is the result you get when you multiply the number by itself twice. For example, the cube of a number [tex]n[/tex] is [tex]n \times n \times n = n^3[/tex]. When the number [tex]n[/tex] is odd, its cube will also be an odd number.
Let's check each option to see if it is the cube of an odd natural number:
6,859: To check whether 6,859 is a cube of an odd number, we need to find an odd integer [tex]n[/tex] such that [tex]n^3 = 6,859[/tex]. Checking through calculations, we find that [tex]19^3 = 19 \times 19 \times 19 = 6,859[/tex]. Therefore, 6,859 is indeed the cube of the odd number 19.
4,913: Checking the cube root of 4,913, we find that [tex]17^3 = 17 \times 17 \times 17 = 4,913[/tex]. Thus, 4,913 is the cube of the odd number 17.
10,648: Checking the cube root of 10,648, calculations show that [tex]22^3 = 22 \times 22 \times 22 = 10,648[/tex]. Since 22 is an even number, 10,648 is the cube of an even number, not an odd number.
2,197: For 2,197, checking its cube root, we find that [tex]13^3 = 13 \times 13 \times 13 = 2,197[/tex]. Thus, 2,197 is the cube of the odd number 13.
Based on our calculations, the numbers that are cubes of odd natural numbers are:
- (a) 6,859
- (b) 4,913
- (d) 2,197
The correct multiple-choice options are (a), (b), and (d).
