To do: We have to find the smallest number by which the given numbers must be divided so that the quotient is a perfect cube. Solution: (i) Prime factorisation of $675=3\times3\times3\times5\times5$ Grouping the factors in triplets of equal factors, we find that $5\times5$ is not a complete triplet. Therefore, dividing $675$ by $5\times5=25$, we get, $675\div25=3\times3\times3\times5\times5\div25$ $=3\times3\times3$ Hence, the smallest number by which the given number must be divided so that the quotient is a perfect cube is 25. (ii) Prime factorisation of $8640=2\times2\times2\times2\times2\times2\times3\times3\times3\times5$ Grouping the factors in triplets of equal factors, we find that $5$ is not a complete triplet. Therefore, dividing $8640$ by $5$, we get, $8640\div5=2\times2\times2\times2\times2\times2\times3\times3\times3\times5\div5$ $=2\times2\times2\times2\times2\times2\times3\times3\times3$ Hence, the smallest number by which the given number must be divided so that the quotient is a perfect cube is 5. (iii) Prime factorisation of $1600=2\times2\times2\times2\times2\times2\times5\times5$ Grouping the factors in triplets of equal factors, we find that $5\times5$ is not a complete triplet. Therefore, dividing $1600$ by $5\times5=25$, we get, $1600\div25=2\times2\times2\times2\times2\times2\times5\times5\div25$ $=2\times2\times2\times2\times2\times2$ Hence, the smallest number by which the given number must be divided so that the quotient is a perfect cube is 25. (iv) Prime factorisation of $8788=2\times2\times13\times13\times13$ Grouping the factors in triplets of equal factors, we find that $2\times2$ is not a complete triplet. Therefore, dividing $8788$ by $2\times2=4$, we get, $8788\div4=2\times2\times13\times13\times13\div4$ $=13\times13\times13$ Hence, the smallest number by which the given number must be divided so that the quotient is a perfect cube is 4. (v) Prime factorisation of $7803=3\times3\times3\times17\times17$ Grouping the factors in triplets of equal factors, we find that $17\times17$ is not a complete triplet. Therefore, dividing $7803$ by $17\times17=289$, we get, $7803\div289=3\times3\times3\times17\times17\div289$ $=3\times3\times3$ Hence, the smallest number by which the given number must be divided so that the quotient is a perfect cube is 289. (vi) Prime factorisation of $107811=3\times3\times3\times3\times11\times11\times11$ Grouping the factors in triplets of equal factors, we find that $3$ is not a complete triplet. Therefore, dividing $107811$ by $3$, we get, $107811\div3=3\times3\times3\times3\times11\times11\times11\div3$ $=3\times3\times3\times11\times11\times11$ Hence, the smallest number by which the given number must be divided so that the quotient is a perfect cube is 3. (vii) Prime factorisation of $35721=3\times3\times3\times3\times3\times3\times7\times7$ Grouping the factors in triplets of equal factors, we find that $7\times7$ is not a complete triplet. Therefore, dividing $35721$ by $7\times7=49$, we get, $35721\div49=3\times3\times3\times3\times3\times3\times7\times7\div49$ $=3\times3\times3\times3\times3\times3$ Hence, the smallest number by which the given number must be divided so that the quotient is a perfect cube is 49. (viii) Prime factorisation of $243000=2\times2\times2\times3\times3\times3\times3\times3\times5\times5\times5$ Grouping the factors in triplets of equal factors, we find that $3\times3$ is not a complete triplet. Therefore, dividing $243000$ by $3\times3=9$, we get, $243000\div9=2\times2\times2\times3\times3\times3\times3\times3\times5\times5\times5\div9$ $=2\times2\times2\times3\times3\times3\times5\times5\times5$ Hence, the smallest number by which the given number must be divided so that the quotient is a perfect cube is 9.  Text Solution Solution : By using prime factorization we could find:<br>(i)`675`<br>`675=3^3times5^2`<br>As we can see that `5^2` is not cubed.<br>Hence, to make the quotient a perfect cube we divide it by `5^2=25`, which gives `27` as quotient where, `27` is a perfect cube. <br><br>(ii)`8640`<br>`8640=2^6times3^3times5`<br>As we can see that `5` is not cubed.<br>Hence, to make the quotient a perfect cube we divide it by `5`, which gives `1728` as quotient where, `1728` is a perfect cube.<br><br>(iii)`1600`<br>`1600=2^6times5^2`<br>As we can see that `5^2` is not cubed.<br>Hence, to make the quotient a perfect cube we divide it by `5^2=25`, which gives `64` as quotient where, `64` is a perfect cube.<br><br>(iv)`8788`<br>`8788=2^2times13^3`<br>As we can see that `2^2` is not cubed.<br>Hence, to make the quotient a perfect cube we divide it by `2^2=4`, which gives `2197` as quotient where, `2197` is a perfect cube.
(i)First find the prime factors of 675 675 = 3 × 3 × 3 × 5 × 5 = 33 × 52 Since 675 is not a perfect cube. To make the quotient a perfect cube we divide it by 52 = 25, which gives 27 as quotient where, 27 is a perfect cube. ∴ 25 is the required smallest number. (ii) 8640 First find the prime factors of 8640 8640 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 5 = 23 × 23 × 33 × 5 Since 8640 is not a perfect cube. To make the quotient a perfect cube we divide it by 5, which gives 1728 as quotient and we know that 1728 is a perfect cube. ∴5 is the required smallest number. (iii) 1600 First find the prime factors of 1600 1600 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 = 23 × 23 × 52 Since 1600 is not a perfect cube. To make the quotient a perfect cube we divide it by 52 = 25, which gives 64 as quotient and we know that 64 is a perfect cube ∴ 25 is the required smallest number. (iv) 8788 First find the prime factors of 8788 8788 = 2 × 2 × 13 × 13 × 13 = 22 × 133 Since 8788 is not a perfect cube. To make the quotient a perfect cube we divide it by 4, which gives 2197 as quotient and we know that 2197 is a perfect cube ∴ 4 is the required smallest number. (v) 7803 First find the prime factors of 7803 7803 = 3 × 3 × 3 × 17 × 17 = 33 × 172 Since 7803 is not a perfect cube. To make the quotient a perfect cube we divide it by 172 = 289, which gives 27 as quotient and we know that 27 is a perfect cube ∴ 289 is the required smallest number. (vi) 107811 First find the prime factors of 107811 107811 = 3 × 3 × 3 × 3 × 11 × 11 × 11 = 33 × 113 × 3 Since 107811 is not a perfect cube. To make the quotient a perfect cube we divide it by 3, which gives 35937 as quotient and we know that 35937 is a perfect cube. ∴ 3 is the required smallest number. (vii) 35721 First find the prime factors of 35721 35721 = 3 × 3 × 3 × 3 × 3 × 3 × 7 × 7 = 33 × 33 × 72 Since 35721 is not a perfect cube. To make the quotient a perfect cube we divide it by 72 = 49, which gives 729 as quotient and we know that 729 is a perfect cube ∴ 49 is the required smallest number. (viii) 243000 First find the prime factors of 243000 243000 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 5 × 5 × 5 = 23 × 33 × 53 × 32 Since 243000 is not a perfect cube. To make the quotient a perfect cube we divide it by 32 = 9, which gives 27000 as quotient and we know that 27000 is a perfect cube ∴ 9 is the required smallest number. We have to find the smallest number by which 675 must be multiplied to obtain a perfect cube Solution 675 On prime factorisation of 675 we get, 675 = 3×3×3×5×5 By assorting the factors in triplets of equal factors, 675 = (3×3×3) x 5×5 Here, 5 cannot be arranged into triplets of equal factors. ∴ We will multiply 675 by 5 to get perfect cube. Answer The smallest number by which 675 must be multiplied to obtain a perfect cube is 5.
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What is the smallest number by which the following number be divided so that quotient is a perfect cube 675?
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