Questions: A Square and a Cube

Q1. What is a perfect square, and how can the units digit help identify one?

Answer (English): A perfect square is the square of a natural number, such as 1, 4, 9 or 16. The final digit of a perfect square is always one of zero, one, four, five, six or nine, so a number ending in 2, 3, 7 or 8 is certainly not a square. The units digit can rule a number out, but it cannot prove that a number is a square; 26 ends in 6 yet is not a square.

Hinglish: Perfect square kisi natural number ka square hota hai, jaise 1, 4, 9, 16. Square ka final digit hamesha zero, one, four, five, six ya nine inmein se hota hai, isliye 2, 3, 7 ya 8 par khatam number pakka square nahi hai. Units digit number ko remove kar sakta hai, par square prove nahi kar sakta; 26 ka last digit 6 hai par woh square nahi hai.

Q2. Why do trailing zeros of a square always come in pairs?

Answer (English): A zero at the end of a number comes from a factor of 10, that is 2 x 5. When a number is squared, each factor of 10 contributes two factors of 10, so the trailing zeros double. For example, 30 ends in one zero and 30^2 = 900 ends in two; 200 ends in two zeros and 200^2 = 40000 ends in four. Since the count doubles, it must be even, so a square cannot end in exactly one or three zeros.

Hinglish: Number ke end ka zero 10 ke factor se aata hai. Square karne par har 10 do 10 ke factors deta hai, isliye trailing zeros double ho jaate hain. Jaise 30^2 = 900 mein do zeros, 200^2 = 40000 mein chaar. Ginti double hoti hai; square ke end mein exactly ek ya teen zeros nahi ho sakte.

Q3. How are odd numbers related to square numbers?

Answer (English): The differences between consecutive squares are consecutive odd numbers: the jump from 1 to 4 is 3, from 4 to 9 it is 5, and from 9 to 16 it is 7. So adding the first n odd numbers gives n^2, and the nth odd number is 2n - 1. This gives a quick way to step from one square to the next, since adding the next odd number produces the next square.

Hinglish: Consecutive squares ke gaps consecutive odd numbers hote hain: 1 se 4 ka jump 3 hai, 4 se 9 ka 5, aur 9 se 16 ka 7. Isliye first n odd numbers ka sum n^2 hota hai, aur nth odd number 2n - 1 hai. Isse ek square se agle square tak jaana aasan hota hai, kyunki agla odd number add karne par agla square milta hai.

Q4. How many numbers lie between 24 squared and 25 squared?

Answer (English): Between n^2 and (n+1)^2 there are 2n numbers. For n = 24 the count is 48. Check: 25^2 - 24^2 - 1 = 625 - 576 - 1 = 48. The two squares are not counted because the question asks for numbers between them.

Hinglish: n^2 aur (n+1)^2 ke beech 2n numbers hote hain. n = 24 ke liye ginti 48 hai. Check: 25^2 - 24^2 - 1 = 625 - 576 - 1 = 48. Dono squares khud count nahi hote kyunki sawaal unke beech ke numbers ka hai.

Q5. How do you find a square root using prime factorisation?

Answer (English): Factorise the number into primes and try to split the factors into two identical groups. If that is possible, the product of either group is the square root. For example, 324 = 2 x 2 x 3 x 3 x 3 x 3 = (2 x 3 x 3) x (2 x 3 x 3), so the square root is 18. If some prime cannot be put into both groups, the number is not a perfect square.

Hinglish: Number ko primes mein factorise karo aur factors ko do identical groups mein baantne ki koshish karo. Agar aisa ho sake to kisi ek group ka product square root hota hai. Jaise 324 = 2 x 2 x 3 x 3 x 3 x 3 = (2 x 3 x 3) x (2 x 3 x 3), isliye square root 18 hai. Agar koi prime dono groups mein nahi aa paaye, to number perfect square nahi hai.

Q6. How can a cube root be estimated without full factorisation?

Answer (English): Use the last digit and the size of the number. The last digit of a cube tells the last digit of the root: a cube ending in 3 comes from a root ending in 7, and one ending in 8 comes from a root ending in 2. Then place the number between two nearby cubes to fix the tens digit. For 4913, the last digit 3 suggests a root ending in 7, and since 4913 lies between 10^3 and 20^3, the root is 17.

Hinglish: Number ke last digit aur size se cube root estimate karo. Cube ka last digit root ka last digit batata hai: 3 par khatam cube ka root 7 par, aur 8 par khatam cube ka root 2 par. Phir number ko do nearby cubes ke beech rakh kar tens digit fix karo. 4913 ka last digit 3 hai, isliye root 7 par khatam hoga; number 10^3 aur 20^3 ke beech hai, isliye root 17 hai.

Q7. Why is 1729 called the Hardy-Ramanujan number?

Answer (English): Hardy visited Ramanujan in hospital and called the taxicab number 1729 dull. Ramanujan replied that 1729 is the smallest number expressible as a sum of two positive cubes in two ways: 1729 = 1^3 + 12^3 = 9^3 + 10^3. Such numbers are called taxicab numbers, and 1729 became famous as the Hardy-Ramanujan number.

Hinglish: Hardy ne hospital mein Ramanujan se kaha ki unki taxicab ka number 1729 dull lagta hai. Ramanujan ne bataya ki 1729 sabse chhota number hai jo do positive cubes ke do alag sums deta hai: 1729 = 1^3 + 12^3 = 9^3 + 10^3. Aise numbers taxicab numbers hain, aur 1729 Hardy-Ramanujan number kehlaya.

Q8. What is the difference between a square and a cube?

Answer (English): A square is the product of a number with itself twice, n x n, written n^2, while a cube multiplies a number by itself three times, n x n x n, written n^3. Squares relate to the area of a square and sums of odd numbers. Cubes relate to the volume of a cube and blocks of consecutive odd numbers. Both have inverse operations: square root and cube root.

Hinglish: Square kisi number ko do baar khud se multiply karne par banta hai, n x n = n^2; cube teen baar multiply karne par, n x n x n = n^3. Squares square ke area aur odd numbers ke sum se jude hain. Cubes cube ke volume aur consecutive odd-number blocks ke sum se jude hain. Dono ke inverse operations square root aur cube root hain.

Q9. A number ends in 5. What can you say about its square?

Answer (English): Its square ends in 25. The squares 5^2 = 25, 15^2 = 225, 25^2 = 625 and 35^2 = 1225 show the pattern. Writing the number as 10a + 5 and expanding gives 100a^2 + 100a + 25, which ends in 25. The digits before 25 come from a(a + 1).

Hinglish: Uska square 25 par khatam hota hai: 5^2 = 25, 15^2 = 225, 25^2 = 625. Number ko 10a + 5 likho aur expand karo: 100a^2 + 100a + 25, jo 25 par khatam hota hai; pehle ke digits a(a + 1) se aate hain.

Q10. Why can a square never end in 7?

Answer (English): A square's last digit depends only on the last digit of the base. The ten possible last digits give endings 0, 1, 4, 9, 6, 5, 6, 9, 4 and 1. The digit 7 never appears, so no perfect square ends in 7; the same rules out 2, 3 and 8.

Hinglish: Square ka last digit sirf number ke last digit par depend karta hai. Das possible last digits ka square karne par endings 0, 1, 4, 9, 6, 5, 6, 9, 4 aur 1 milte hain. Digit 7 kabhi nahi aata, isliye koi perfect square 7 par khatam nahi ho sakta; isi tarah 2, 3 aur 8 bhi rule out ho jaate hain.

Quiz below: squares, roots and cubes. Review anything shaky.