Answer :
Let's determine the smallest number that when multiplied with each of the given numbers results in a perfect cube. A perfect cube is a number which is the cube of an integer. We need to check the prime factorization of each number and find what needs to be added in the prime factorization to make the powers of all primes in the factorization a multiple of 3.
### (a) Number: 4
- Prime factorization of [tex]\( 4 \)[/tex]: [tex]\( 2^2 \)[/tex]
- To make it a perfect cube, the power of 2 should be a multiple of 3.
- [tex]\( 2^2 \)[/tex] needs to be multiplied by [tex]\( 2^1 \)[/tex], making it [tex]\( 2^3 = 8 \)[/tex].
Thus, the smallest number is [tex]\( 8 \)[/tex].
### (b) Number: 25
- Prime factorization of [tex]\( 25 \)[/tex]: [tex]\( 5^2 \)[/tex]
- To make it a perfect cube, the power of 5 should be a multiple of 3.
- [tex]\( 5^2 \)[/tex] needs to be multiplied by [tex]\( 5^1 \)[/tex], making it [tex]\( 5^3 = 125 \)[/tex].
Thus, the smallest number is [tex]\( 125 \)[/tex].
### (c) Number: 72
- Prime factorization of [tex]\( 72 \)[/tex]: [tex]\( 2^3 \times 3^2 \)[/tex]
- To make it a perfect cube, the power of each prime should be a multiple of 3.
- The [tex]\( 2^3 \)[/tex] part is already a perfect cube.
- [tex]\( 3^2 \)[/tex] needs to be multiplied by [tex]\( 3^1 \)[/tex], making it [tex]\( 3^3 \)[/tex].
Thus, the smallest number is [tex]\( 3 = 72 \times 3 = 216 \)[/tex].
### (d) Number: 169
- Prime factorization of [tex]\( 169 \)[/tex]: [tex]\( 13^2 \)[/tex]
- To make it a perfect cube, the power of 13 should be a multiple of 3.
- [tex]\( 13^2 \)[/tex] needs to be multiplied by [tex]\( 13^1 \)[/tex], making it [tex]\( 13^3 = 2197 \)[/tex].
Thus, the smallest number is [tex]\( 2197 \)[/tex].
### (e) Number: 256
- Prime factorization of [tex]\( 256 \)[/tex]: [tex]\( 2^8 \)[/tex]
- To make it a perfect cube, the power of 2 should be a multiple of 3.
- [tex]\( 2^8 \)[/tex] needs to be multiplied by [tex]\( 2^1 \)[/tex], making it [tex]\( 2^9 = 512 \)[/tex].
Thus, the smallest number is [tex]\( 512 \)[/tex].
### (f) Number: 484
- Prime factorization of [tex]\( 484 \)[/tex]: [tex]\( 2^2 \times 11^2 \)[/tex]
- To make it a perfect cube, the power of each prime should be a multiple of 3.
- [tex]\( 2^2 \)[/tex] needs to be multiplied by [tex]\( 2^1 \)[/tex], making it [tex]\( 2^3 \)[/tex].
- [tex]\( 11^2 \)[/tex] needs to be multiplied by [tex]\( 11^1 \)[/tex], making it [tex]\( 11^3 \)[/tex].
Thus, the smallest number is [tex]\( 2 \times 11 = 22 \)[/tex]. Hence, [tex]\( 484 \times 22 = 10648 \)[/tex].
### (g) Number: 968
- Prime factorization of [tex]\( 968 \)[/tex]: [tex]\( 2^3 \times 11^1 \)[/tex]
- To make it a perfect cube, the power of each prime should be a multiple of 3.
- The [tex]\( 2^3 \)[/tex] part is already a perfect cube.
- [tex]\( 11^1 \)[/tex] needs to be multiplied by [tex]\( 11^2 \)[/tex], making it [tex]\( 11^3 \)[/tex].
Thus, the smallest number is [tex]\( 11^2 = 121 \)[/tex]. Hence, [tex]\( 968 \times 121 = 10648 \)[/tex].
### (h) Number: 8788
- Prime factorization of [tex]\( 8788 \)[/tex]: [tex]\( 2^2 \times 37^1 \times 59^1 \)[/tex]
- To make it a perfect cube, the power of each prime should be a multiple of 3.
- [tex]\( 2^2 \)[/tex] needs to be multiplied by [tex]\( 2^1 \)[/tex], making it [tex]\( 2^3 \)[/tex].
- [tex]\( 37^1 \)[/tex] needs to be multiplied by [tex]\( 37^2 \)[/tex], making it [tex]\( 37^3 \)[/tex].
- [tex]\( 59^1 \)[/tex] needs to be multiplied by [tex]\( 59^2 \)[/tex], making it [tex]\( 59^3 \)[/tex].
Thus, the smallest number is [tex]\( 2 \times 37^2 \times 59^2 = 2 \times 1369 \times 3481 = 95676 / 5 = 17576 \)[/tex]. Hence, [tex]\( 8788 \times 2 \times 1369 \times 3481 = 17576 \)[/tex].
So, all required numbers are:
```
(a) 8
(b) 125
(c) 216
(d) 2197
(e) 512
(f) 10648
(g) 10648
(h) 17576
```
### (a) Number: 4
- Prime factorization of [tex]\( 4 \)[/tex]: [tex]\( 2^2 \)[/tex]
- To make it a perfect cube, the power of 2 should be a multiple of 3.
- [tex]\( 2^2 \)[/tex] needs to be multiplied by [tex]\( 2^1 \)[/tex], making it [tex]\( 2^3 = 8 \)[/tex].
Thus, the smallest number is [tex]\( 8 \)[/tex].
### (b) Number: 25
- Prime factorization of [tex]\( 25 \)[/tex]: [tex]\( 5^2 \)[/tex]
- To make it a perfect cube, the power of 5 should be a multiple of 3.
- [tex]\( 5^2 \)[/tex] needs to be multiplied by [tex]\( 5^1 \)[/tex], making it [tex]\( 5^3 = 125 \)[/tex].
Thus, the smallest number is [tex]\( 125 \)[/tex].
### (c) Number: 72
- Prime factorization of [tex]\( 72 \)[/tex]: [tex]\( 2^3 \times 3^2 \)[/tex]
- To make it a perfect cube, the power of each prime should be a multiple of 3.
- The [tex]\( 2^3 \)[/tex] part is already a perfect cube.
- [tex]\( 3^2 \)[/tex] needs to be multiplied by [tex]\( 3^1 \)[/tex], making it [tex]\( 3^3 \)[/tex].
Thus, the smallest number is [tex]\( 3 = 72 \times 3 = 216 \)[/tex].
### (d) Number: 169
- Prime factorization of [tex]\( 169 \)[/tex]: [tex]\( 13^2 \)[/tex]
- To make it a perfect cube, the power of 13 should be a multiple of 3.
- [tex]\( 13^2 \)[/tex] needs to be multiplied by [tex]\( 13^1 \)[/tex], making it [tex]\( 13^3 = 2197 \)[/tex].
Thus, the smallest number is [tex]\( 2197 \)[/tex].
### (e) Number: 256
- Prime factorization of [tex]\( 256 \)[/tex]: [tex]\( 2^8 \)[/tex]
- To make it a perfect cube, the power of 2 should be a multiple of 3.
- [tex]\( 2^8 \)[/tex] needs to be multiplied by [tex]\( 2^1 \)[/tex], making it [tex]\( 2^9 = 512 \)[/tex].
Thus, the smallest number is [tex]\( 512 \)[/tex].
### (f) Number: 484
- Prime factorization of [tex]\( 484 \)[/tex]: [tex]\( 2^2 \times 11^2 \)[/tex]
- To make it a perfect cube, the power of each prime should be a multiple of 3.
- [tex]\( 2^2 \)[/tex] needs to be multiplied by [tex]\( 2^1 \)[/tex], making it [tex]\( 2^3 \)[/tex].
- [tex]\( 11^2 \)[/tex] needs to be multiplied by [tex]\( 11^1 \)[/tex], making it [tex]\( 11^3 \)[/tex].
Thus, the smallest number is [tex]\( 2 \times 11 = 22 \)[/tex]. Hence, [tex]\( 484 \times 22 = 10648 \)[/tex].
### (g) Number: 968
- Prime factorization of [tex]\( 968 \)[/tex]: [tex]\( 2^3 \times 11^1 \)[/tex]
- To make it a perfect cube, the power of each prime should be a multiple of 3.
- The [tex]\( 2^3 \)[/tex] part is already a perfect cube.
- [tex]\( 11^1 \)[/tex] needs to be multiplied by [tex]\( 11^2 \)[/tex], making it [tex]\( 11^3 \)[/tex].
Thus, the smallest number is [tex]\( 11^2 = 121 \)[/tex]. Hence, [tex]\( 968 \times 121 = 10648 \)[/tex].
### (h) Number: 8788
- Prime factorization of [tex]\( 8788 \)[/tex]: [tex]\( 2^2 \times 37^1 \times 59^1 \)[/tex]
- To make it a perfect cube, the power of each prime should be a multiple of 3.
- [tex]\( 2^2 \)[/tex] needs to be multiplied by [tex]\( 2^1 \)[/tex], making it [tex]\( 2^3 \)[/tex].
- [tex]\( 37^1 \)[/tex] needs to be multiplied by [tex]\( 37^2 \)[/tex], making it [tex]\( 37^3 \)[/tex].
- [tex]\( 59^1 \)[/tex] needs to be multiplied by [tex]\( 59^2 \)[/tex], making it [tex]\( 59^3 \)[/tex].
Thus, the smallest number is [tex]\( 2 \times 37^2 \times 59^2 = 2 \times 1369 \times 3481 = 95676 / 5 = 17576 \)[/tex]. Hence, [tex]\( 8788 \times 2 \times 1369 \times 3481 = 17576 \)[/tex].
So, all required numbers are:
```
(a) 8
(b) 125
(c) 216
(d) 2197
(e) 512
(f) 10648
(g) 10648
(h) 17576
```
