Questions: Number Play
Number Play24 min readLesson 7 of 7
Q1 What is the difference between a multiple and a factor?
Answer (English): A multiple of a number is the product of that number and a whole number, such as 6, 12, 18 for the number 6. A factor divides a number exactly, leaving no remainder, such as 2 and 9 for 18. Every number is both a factor and a multiple of itself.
Hinglish: Multiple kisi number ko whole number se multiply karke milta hai, jaise 6 ke multiples 6, 12, 18. Factor woh number hai jo bina remainder ke divide kare, jaise 18 ke factors 2 aur 9. Har number khud ka factor aur multiple dono hota hai.
Q2 Why is the sum of four consecutive numbers always even?
Answer (English): If the numbers are n, n + 1, n + 2 and n + 3, their sum is 4n + 6 = 2(2n + 3). The factor 2 shows the sum is always even, whatever whole number n is chosen.
Hinglish: Chaar consecutive numbers n, n+1, n+2, n+3 lo to unka sum 4n + 6 = 2(2n + 3) hota hai. Isme 2 ka factor hai, isliye n koi bhi ho sum hamesha even hota hai.
Q3 Why does a number and the sum of its digits have the same remainder when divided by 9?
Answer (English): Each place value 10, 100, 1000 is one more than a multiple of 9 (9, 99, 999). Expanding the number shows it equals a multiple of 9 plus the sum of its digits, so the remainder on division by 9 comes from the digit sum alone.
Hinglish: Har place value 10, 100, 1000, 9 ke kisi multiple se ek zyada hai (9, 99, 999). Number ko expand karo to woh 9 ke multiple plus digits ke sum ke barabar hota hai, isliye 9 se remainder sirf digit sum se aata hai.
Q4 How do you test divisibility by 11?
Answer (English): Add the digits in alternating positions starting from the units digit: plus, minus, plus, minus. If the resulting difference is 0, 11 or a multiple of 11, the number is divisible by 11. Otherwise the difference gives the remainder after adjusting it to a positive value.
Hinglish: Units digit se shuru karke alternate digits par plus aur minus lagao. Agar difference 0, 11 ya 11 ka multiple ho to number 11 se divisible hai. Warna difference ko positive banakar remainder mil jata hai.
Q5 What is a digital root, and how does it relate to multiples of 9?
Answer (English): The digital root is the single digit obtained by repeatedly adding the digits of a number. It equals the remainder on division by 9, with the convention that a remainder of 0 is written as 9. So every multiple of 9, and only those numbers, has digital root 9.
Hinglish: Digital root digits ko baar-baar jodkar milne wala single digit hai. Woh 9 se division ka remainder hota hai, aur remainder 0 ho to use 9 likhte hain. Isliye 9 ke har multiple ka digital root 9 hota hai aur baaki numbers ka nahi.
Q6 Under what condition does adding two multiples of 3 produce a result divisible by 6?
Answer (English): Write the multiples as 3m and 3n. Their sum is 3(m + n), which is a multiple of 6 exactly when m + n is even, that is, when m and n have the same parity. Two even multiples of 3 and two odd multiples of 3 both work; one even and one odd do not.
Hinglish: Multiples ko 3m aur 3n likho. Sum 3(m + n) hota hai, jo 6 ka multiple tabhi hai jab m + n even ho, yani m aur n ki parity same ho. Do even multiples of 3 aur do odd multiples of 3 dono ka sum multiple of 6 hota hai; ek even ek odd ka nahi.
Q7 How do you solve a cryptarithm?
Answer (English): Read the units column first and list possible digits and carries, then move to the tens and higher columns. Different letters stand for different digits, and a number's leading digit cannot be zero. Check the complete solution by substituting the digits back into the sum or product.
Hinglish: Pehle units column padho, possible digits aur carry list karo, phir tens aur aage ke columns solve karo. Alag letters alag digits hote hain aur kisi number ka pehla digit 0 nahi hota. Aakhir mein digits substitute karke answer check karo.
Q8 If 31z5 is a multiple of 9, why are there two possible values of z?
Answer (English): The digit sum is 9 + z. This must be a multiple of 9, and with z a single digit the only possibilities are 9 and 18. So z = 0 or z = 9, giving 3105 and 3195. There are two answers because the digit sum can reach either of two multiples of 9.
Hinglish: Digits ka sum 9 + z hai. Ise 9 ka multiple hona chahiye, aur z single digit hai, isliye sum 9 ya 18 hi ho sakta hai. Isliye z = 0 ya 9, aur numbers 3105 aur 3195 hain. Do answers isliye hain kyunki digit sum 9 ke do multiples tak pahunch sakta hai.
Q9 When is a statement about numbers 'sometimes true' rather than always or never true?
Answer (English): A statement is sometimes true when at least one example makes it true and at least one counterexample makes it false. The chapter uses algebra to show that the statement's truth depends on a hidden condition, such as the parity of the multipliers in 3m + 3n.
Hinglish: Statement sometimes true tab hota hai jab kam se kam ek example usse sach banaye aur kam se kam ek counterexample jhoota banaye. Chapter mein algebra se dikhate hain ki sach hona kisi chhupi condition par depend karta hai, jaise 3m + 3n mein m aur n ki parity.
Q10 Why must a multiple of 18 be divisible by both 2 and 9?
Answer (English): 18 = 2 × 9, so any multiple of 18 is 18k = 2(9k), which is even and divisible by 9. The two conditions are necessary and together sufficient because 2 and 9 are coprime; their LCM is 18.
Hinglish: 18 = 2 × 9 hai, isliye 18 ka multiple 18k = 2(9k) hota hai, jo even bhi hai aur 9 se divisible bhi. Dono conditions zaroori hain aur saath mein sufficient bhi, kyunki 2 aur 9 coprime hain aur unka LCM 18 hai.
The five-question quiz for this chapter is separate. Take it without scrolling back to these answers, then revisit any question you missed. For divisibility questions, practise writing the digit sum or alternating sum as part of the answer, because the working earns marks even when the final digit is wrong.