Number System & Arithmetic Simplification — JNV Class 6 PYQs: 20 flashcards
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1. What is the divisibility rule for 9?
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A number is divisible by 9 if and only if the sum of its digits is divisible by 9.
2. If a two-digit number has digit sum 8, what form can it be written in?
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It can be written as $9k + 8$ for some integer $k$.
3. What happens to the remainder mod 9 when you subtract a multiple of 9 from a number?
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The remainder mod 9 stays the same.
4. List all two-digit numbers whose digits sum to 8.
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17, 26, 35, 44, 53, 62, 71, 80.
5. What is the remainder when a number with digit sum 8 is divided by 9?
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The remainder is 8.
6. If you subtract 18 from a number with digit sum 8, what is the remainder mod 9 of the result?
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The remainder is still 8.
7. Why does subtracting 18 not change the digit sum mod 9?
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Because 18 is a multiple of 9, and subtracting a multiple of 9 preserves the remainder mod 9.
8. What is the intended answer to the JNV question about digit sum 8 and subtracting 18?
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The intended answer is 9.
9. Is the result of subtracting 18 from a number with digit sum 8 always divisible by 9?
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No, it is never divisible by 9; it always leaves remainder 8.
10. What is the correct precise statement about the result of subtracting 18 from a number with digit sum 8?
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The result always leaves remainder 8 when divided by 9.
11. Show that 35 − 18 is not divisible by 2, 5, or 8.
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35 − 18 = 17, which is odd (not divisible by 2), does not end in 0 or 5 (not divisible by 5), and 17 is not a multiple of 8.
12. Show that 44 − 18 is not divisible by 5 or 8.
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44 − 18 = 26, which does not end in 0 or 5, and 26 is not a multiple of 8.
13. What is the common mistake students make with this problem?
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They assume 'divisible by 9' means the result is a multiple of 9, but actually the result leaves remainder 8.
14. How can you quickly find all two-digit numbers with digit sum 8?
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Start with 17 and keep adding 9: 17, 26, 35, 44, 53, 62, 71, 80.
15. What does the digit-sum rule say about a number and its remainder mod 9?
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A number is congruent to the sum of its digits modulo 9.
16. If a number is $9k+8$, what is its digit sum mod 9?
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The digit sum mod 9 is 8.
17. Why is 18 called a 'magic number' in this problem?
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Because 18 is a multiple of 9, so subtracting it does not change the remainder mod 9.
18. What should you think when you see 'digit sum' and a subtracted multiple of 9?
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Think of the divisibility rule for 9.
19. What is the remainder when 80 is divided by 9?
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8, because 80 = 9 × 8 + 8.
20. What is the remainder when 17 is divided by 9?
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8, because 17 = 9 × 1 + 8.
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