Number System & Arithmetic Simplification — JNV Class 6 PYQs: 16 MCQs with answers
All 16 multiple-choice questions from this kit, free and with no sign-up.
Warm-up · 5 questions
Q1. A two-digit number has digit sum 8. Which of the following could be the number?
BeginnerShow the answer
D · 35
3 + 5 = 8, so 35 has digit sum 8. The others have digit sums 9, 10, and 9 respectively.
Q2. What is the result of subtracting 18 from 53?
BeginnerShow the answer
B · 35
53 − 18 = 35.
Q3. Which of the following numbers is divisible by 9?
BeginnerShow the answer
C · 27
27 has digit sum 9, so it is divisible by 9. The others have digit sum 8.
Q4. A number is divisible by 9 if and only if the sum of its digits is divisible by 9.
BeginnerShow the answer
A · True
A number is divisible by 9 if and only if the sum of its digits is divisible by 9.
Q5. When a two-digit number with a digit sum of 8 has 18 subtracted from it, what is the remainder when the result is divided by 9?
BeginnerShow the answer
D · 8
The correct answer is 8 because subtracting 18 from a number with a digit sum of 8 preserves the remainder when divided by 9. Thus, the result always leaves a remainder of 8.
Building up · 5 questions
Q6. Take the number 44 (digit sum 8). Subtract 18. Is the result divisible by 5?
IntermediateShow the answer
B · No, because 26 does not end in 0 or 5.
44 − 18 = 26. A number is divisible by 5 only if it ends in 0 or 5. 26 ends in 6, so it is not divisible by 5.
Q7. Which of the following statements is true about the result of subtracting 18 from a two-digit number with digit sum 8?
IntermediateShow the answer
D · It always leaves remainder 8 when divided by 9.
The result is of the form $9m+8$, so it leaves remainder 8 mod 9. It is not always divisible by 2 or 5 (e.g., 35−18=17).
Q8. Why does subtracting 18 from a number with digit sum 8 not change the remainder mod 9?
IntermediateShow the answer
A · Because 18 is a multiple of 9.
Subtracting a multiple of 9 preserves the remainder modulo 9. 18 = 2×9, so it is a multiple of 9.
Q9. Consider the number 62 (digit sum 8). After subtracting 18, what is the result and is it divisible by 8?
IntermediateShow the answer
B · 44, not divisible by 8
62 − 18 = 44. 44 ÷ 8 = 5.5, so not divisible by 8.
Q10. Which of the following is the correct expression for a two-digit number with digit sum 8?
IntermediateShow the answer
C · $9k + 8$
A number with digit sum 8 leaves remainder 8 mod 9, so it can be written as $9k+8$.
Topper level · 6 questions
Q11. A student says: 'If a number has digit sum 8, then subtracting 18 makes it divisible by 9.' What is wrong with this reasoning?
AdvancedShow the answer
C · Subtracting 18 preserves the remainder mod 9, so the result still leaves remainder 8, not 0.
The student mistakenly thinks subtracting 18 makes it divisible by 9. Actually, since 18 is a multiple of 9, the remainder mod 9 stays 8, so the result is never divisible by 9.
Q12. Which of the following numbers, when subtracted by 18, gives a number divisible by 2?
AdvancedShow the answer
B · 26
26 − 18 = 8, which is even and divisible by 2. 35−18=17 (odd), 53−18=35 (odd), 71−18=53 (odd).
Q13. Assertion (A): When you subtract 18 from a two-digit number with digit sum 8, the result always leaves remainder 8 when divided by 9. Reason (R): 18 is a multiple of 9, and subtracting a multiple of 9 does not change the remainder modulo 9. Which of the following is correct?
AdvancedShow the answer
B · Both A and R are true, and R is the correct explanation of A.
A is true: the result is $9m+8$, so remainder 8. R is true: subtracting a multiple of 9 preserves remainder mod 9. R explains A because 18 is a multiple of 9.
Q14. Which of the following is the intended answer to the JNV question: 'A two-digit number has digit sum 8. Subtract 18. The result is always divisible by which number?'
AdvancedShow the answer
C · 9
The intended answer is 9 because the digit-sum rule for 9 is the key. Although the result is not actually divisible by 9, the exam key expects 9.
Q15. Which of the following numbers, when subtracted by 18, gives a result that is divisible by 8?
AdvancedShow the answer
B · 26
26 − 18 = 8, which is divisible by 8. 35−18=17 (not divisible by 8), 44−18=26 (not divisible by 8), 53−18=35 (not divisible by 8).
Q16. Statement I: A number with digit sum 8 leaves remainder 8 when divided by 9. Statement II: Subtracting 18 from such a number changes the remainder mod 9. Statement III: The result of subtracting 18 from such a number always leaves remainder 8 when divided by 9. Which of the following is correct?
AdvancedShow the answer
C · I and III only
I is true. II is false because subtracting 18 (a multiple of 9) does not change the remainder mod 9. III is true. So I and III only.
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