was 1729. While talking to Ramanujan, Hardy described this number | Class 8 Mathematics Notes
By ConceptScroll Team · Published on 17 July 2026 · 4 min read
was 1729. While talking to Ramanujan, Hardy described this number – this guide gives you a concise, exam-ready overview of was 1729. While talking to Ramanujan, Hardy described this number from Class 8 Mathematics, written by ConceptScroll editors and reviewed against the latest NCERT textbook.
Properties of Cubes
This section delves into the properties of cubes of numbers. It explains that the cube of a positive integer is always positive and increases rapidly as the number increases. The section also discusses the behavior of cubes of negative numbers, which are negative because the cube of a negative number is negative (since multiplying three negative numbers results in a negative product). The section highlights that cubes are used in various mathematical contexts such as volume calculation, algebraic identities, and number theory. It also introduces the algebraic identity for the sum of cubes: a³ + b³ = (a + b)(a² - ab + b²), which is useful for factorization and solving equations. Understanding these properties helps students grasp the significance of the sum of cubes in the context of the number 1729.
📊 Diagram: A flowchart showing the factorization of sum of cubes using the identity with example values a=1 and b=12.
🧪 Activity: Students are encouraged to factorize sums of cubes using the identity and verify results by multiplication.
🔗 Connection: This section provides the algebraic tools to understand the structure of numbers like 1729 and leads to exploring more about taxicab numbers and their significance.
Frequently asked questions
1. Which of the following numbers are not perfect cubes? (i) 216 (ii) 128 (iii) 1000 (iv) 100 (v) 46656
A perfect cube is a number that can be expressed as the cube of an integer.
(i) 216 = 6^3, so it is a perfect cube. (ii) 128 = 2^7, not a perfect cube because the exponent 7 is not a multiple of 3. (iii) 1000 = 10^3, so it is a perfect cube. (iv) 100 is not a perfect cube because it cannot be expressed as n^3 for any integer n. (v) 46656 = 36^3, so it is a perfect cube.
Therefore, numbers which are not perfect cubes are 128 and 100.
2. Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube. (i) 243 (ii) 256 (iii) 72 (iv) 675 (v) 100
To find the smallest number to multiply so that the product is a perfect cube, prime factorize each number and adjust the powers to multiples of 3.
(i) 243 = 3^5 To make power of 3 a multiple of 3, multiply by 3^(3 - (5 mod 3)) = 3^(3 - 2) = 3^1 = 3 Answer: 3
(ii) 256 = 2^8 Next multiple of 3 after 8 is 9, so multiply by 2^(9-8) = 2^1 = 2 Answer: 2
(iii) 72 = 2^3 × 3^2 3^2 needs one more 3 to make power 3, multiply by 3 Answer: 3
(iv) 675 = 3^3 × 5^2 5^2 needs one more 5 to make power 3, mul
3. Find the smallest number by which each of the following numbers must be divided to obtain a perfect cube. (i) 81 (ii) 128 (iii) 135 (iv) 192 (v) 704
To find the smallest number to divide so that the quotient is a perfect cube, prime factorize each number and remove the excess powers to make powers multiples of 3.
(i) 81 = 3^4 Remove one 3 to get 3^3, so divide by 3 Answer: 3
(ii) 128 = 2^7 Remove one 2 to get 2^6 (6 is multiple of 3), so divide by 2 Answer: 2
(iii) 135 = 3^3 × 5^1 5^1 is not multiple of 3, remove 5 to get 3^3 Divide by 5 Answer: 5
(iv) 192 = 2^6 × 3^1 3^1 is not multiple of 3, remove 3 Divide by 3 Answer: 3
(v) 704 = 2^
4. Parikshit makes a cuboid of plasticine of sides 5cm, 2cm, 5cm. How many such cuboids will he need to form a cube?
Volume of one cuboid = 5 × 2 × 5 = 50 cm³ Let the side of the cube be 'a' cm. Since the cube is formed by joining these cuboids, the volume of the cube must be a perfect cube and a multiple of 50. Find the smallest cube volume divisible by 50. Prime factorize 50 = 2 × 5² To make a perfect cube, powers of prime factors must be multiples of 3. For 2: power is 1, needs 2 more → 2^3 For 5: power is 2, needs 1 more → 5^3 So cube volume = 2^3 × 5^3 = 8 × 125 = 1000 cm³ Number of cuboids = Total volume
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