If a number leaves remainder (6) when divided by (7), what happens after adding (1)?
Correct answer: A
Step 1: The original number is (7q+6). Step 2: Adding (1) gives (7q+7=7(q+1)). Step 3: Therefore, the new number is exactly divisible by (7).
Mathematics
Euclid’s Division Lemma
यूक्लिड का विभाजन प्रमेय
In Class 10 Mathematics, under the chapter Real Numbers, Euclid’s Division Lemma introduces the relationship a = bq + r, where a and b are positive integers, q is the quotient, and the remainder r satisfies 0 ≤ r < b. Students learn how repeated division forms Euclid’s division algorithm and use it to find the highest common factor (HCF) of two numbers. The topic also strengthens understanding of divisibility, quotients, remainders, and the logical steps used in number-theory proofs.
TOPIC PRACTICE
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Easy · Level 6
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- The new number will be exactly divisible by (7)
- The new number will still leave remainder (6)
- The new number will leave remainder (7)
- The new number will be less than (7)
Easy · Level 6
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- (0)
- (1)
- (8)
- (9)
Easy · Level 6
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- (q=8,\ r=3)
- (q=9,\ r=-1)
- (q=7,\ r=7)
- (q=8,\ r=4)
Easy · Level 6
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- The remainder is less than the divisor
- The remainder is greater than the divisor
- The remainder is always (1)
- The quotient is always equal to the divisor
Easy · Level 6
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- (a) is exactly divisible by (b)
- (a) is less than (b)
- (r) is greater than (b)
- (q) is always (0)
Easy · Level 6
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- (1)
- (14)
- (7)
- (0)
Easy · Level 6
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- (7)
- (8)
- (6)
- (14)
Easy · Level 6
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- (29)
- (24)
- (36)
- (48)
Easy · Level 6
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- (45)
- (42)
- (49)
- (56)
Easy · Level 6
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- (0)
- (1)
- (5)
- (9)
Easy · Level 6
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- The quotient is (6) and the remainder is (6)
- The quotient is (13) and the remainder is (6)
- The quotient is (6) and the remainder is (13)
- The quotient is (84) and the remainder is (6)
Easy · Level 6
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- (16)
- (15)
- (17)
- (1)
Easy · Level 6
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- (a<b)
- (a=b)
- (a>b)
- (a=0)
Easy · Level 6
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- (5=9 \times 0+5)
- (5=9 \times 1-4)
- (5=9 \times 1+5)
- (5=9 \times 0+9)
Easy · Level 6
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- Exactly divisible by (3)
- Leaves remainder (2) on division by (3)
- Leaves remainder (3) on division by (3)
- Always less than (3)
Easy · Level 6
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- (1)
- (10)
- (11)
- (0)
Easy · Level 6
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- (12)
- (11)
- (13)
- (1)
Easy · Level 6
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- (70=9 \times 7+7)
- (70=9 \times 6+16)
- (70=9 \times 8-2)
- (70=9 \times 5+25)
Easy · Level 6
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- (64=6 \times 10+4)
- (64=6 \times 9+10)
- (64=6 \times 11-2)
- (64=6 \times 8+16)
Easy · Level 6
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- (0)
- (10)
- (11)
- (1)
Easy · Level 6
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- The remainder must be less than the divisor
- The remainder is always equal to the divisor
- The quotient is always (21)
- The dividend is always (0)
Easy · Level 6
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- Quotient (9), remainder (6)
- Quotient (16), remainder (6)
- Quotient (6), remainder (9)
- Quotient (150), remainder (16)
Easy · Level 6
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- (0)
- (1)
- (4)
- (5)
Easy · Level 6
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- (73=12 \times 6+1)
- (73=12 \times 5+13)
- (73=12 \times 7-11)
- (73=12 \times 6+12)
Easy · Level 6
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- (0)
- (1)
- (a)
- (q)
If a number leaves remainder (8) when divided by (9), what will be the new remainder after adding (1)?
Correct answer: A
Step 1: The number is of the form (9q+8). Step 2: Adding (1) gives (9q+9=9(q+1)). Step 3: Therefore, the new remainder is (0).
If (35) is divided by (4), what will be the quotient and remainder?
Correct answer: A
Step 1: (4 \times 8=32) and (4 \times 9=36). Step 2: (36) is greater than (35), so (q=8). Step 3: (35-32=3), so (r=3).
Which statement is always true according to Euclid’s Division Lemma?
Correct answer: A
Step 1: The lemma gives the remainder range (0 \le r < b). Step 2: This means the remainder is always less than the divisor. Step 3: In statement-based questions, this rule is very useful.
If (r=0) in (a=bq+r), which conclusion is correct?
Correct answer: A
Step 1: (r=0) means nothing is left after division. Step 2: Then (a=bq), so (a) is exactly divisible by (b). Step 3: Use zero remainder to identify divisibility.
What is the remainder when (99) is divided by (14)?
Correct answer: A
Step 1: (14 \times 7=98). Step 2: (99-98=1), so the remainder is (1). Step 3: The nearest smaller multiple helps find the remainder quickly.
What is the quotient when (99) is divided by (14)?
Correct answer: A
Step 1: (14 \times 7=98) and (14 \times 8=112). Step 2: (112) is greater than (99), so the quotient is (7). Step 3: For the quotient, choose a multiple less than or equal to the dividend.
Which number can leave remainder (5) when divided by (12)?
Correct answer: A
Step 1: For remainder (5), the number should be of the form (12q+5). Step 2: (29=12 \times 2+5), so its remainder is (5). Step 3: Check each option by division.
Which number will leave remainder (3) when divided by (7)?
Correct answer: A
Step 1: To leave remainder (3) on division by (7), the form should be (7q+3). Step 2: (45=7 \times 6+3). Step 3: The remainder is less than the divisor, so (45) is correct.
If a number leaves remainder (0) when divided by (10), what will its units digit be?
Correct answer: A
Step 1: A number exactly divisible by (10) is of the form (10q). Step 2: Such a number has units digit (0). Step 3: In division by (10), use the units digit for a quick decision.
If (84=13 \times 6+6), which statement is correct?
Correct answer: A
Step 1: Compare with (a=bq+r). Step 2: In (84=13 \times 6+6), (q=6) and (r=6). Step 3: Remainder (6) is less than divisor (13), so the form is correct.
If a number is divided by (16), how many possible remainders can there be?
Correct answer: A
Step 1: On division by (16), remainders can be from (0) to (15). Step 2: These are (16) values in total. Step 3: For a divisor (b), there are (b) possible remainders.
If (q=0) and (r=a) in (a=bq+r), which statement about (a) and (b) is correct?
Correct answer: A
Step 1: Putting (q=0) gives (a=b \times 0+r), so (a=r). Step 2: Since (r<b) is necessary, (a<b). Step 3: When the dividend is smaller than the divisor, the quotient can be (0).
What is the Euclidean form when (5) is divided by (9)?
Correct answer: A
Step 1: The dividend (5) is smaller than the divisor (9). Step 2: So the quotient is (0) and the remainder is (5). Step 3: Note that remainder (5) is less than (9).
If a number leaves remainder (2) when divided by (3), what will the new number be after adding (1)?
Correct answer: A
Step 1: The original number is (3q+2). Step 2: Adding (1) gives (3q+3=3(q+1)). Step 3: Therefore, the new number is exactly divisible by (3).
What is the remainder when (121) is divided by (10)?
Correct answer: A
Step 1: (10 \times 12=120). Step 2: (121-120=1), so the remainder is (1). Step 3: In division by (10), the units digit helps find the remainder.
What is the quotient when (121) is divided by (10)?
Correct answer: A
Step 1: (10 \times 12=120) and (10 \times 13=130). Step 2: (130) is greater than (121), so the quotient is (12). Step 3: For the quotient, always take the nearest smaller multiple.
In which option is the remainder in the correct range for (a=bq+r)?
Correct answer: A
Step 1: The correct remainder must be at least (0) and less than (9). Step 2: In (70=9 \times 7+7), the remainder is (7), which is less than (9). Step 3: Negative or too large remainders are not accepted.
If (a=64) and (b=6), which is the correct form of (a=bq+r)?
Correct answer: A
Step 1: (6 \times 10=60) and (6 \times 11=66). Step 2: (60) is the correct smaller multiple and (64-60=4). Step 3: Since (4<6), (64=6 \times 10+4) is the correct form.
If a number leaves remainder (10) when divided by (11), what will be the new remainder after adding (1)?
Correct answer: A
Step 1: The number is of the form (11q+10). Step 2: Adding (1) gives (11q+11=11(q+1)). Step 3: Therefore, the new remainder is (0).
A number is said to leave remainder (21) when divided by (21). Why is this wrong?
Correct answer: A
Step 1: The remainder range is (0 \le r < b). Step 2: Here (r=21) and (b=21), so the condition (r<b) is not satisfied. Step 3: A remainder is never written equal to the divisor.
If (150=16 \times 9+6), what are the quotient and remainder?
Correct answer: A
Step 1: Compare with the form (a=bq+r). Step 2: (16) is the divisor, (9) is the quotient, and (6) is the remainder. Step 3: Remainder (6) is less than (16), so the form is correct.
If a number leaves remainder (1) when divided by (5), what will be the new remainder after adding (4)?
Correct answer: A
Step 1: The original number is (5q+1). Step 2: Adding (4) gives (5q+5=5(q+1)). Step 3: The new number is exactly divisible by (5), so the remainder is (0).
Which is the correct Euclidean form when (73) is divided by (12)?
Correct answer: A
Step 1: (12 \times 6=72) and (12 \times 7=84). Step 2: (72) is the correct smaller multiple and (73-72=1). Step 3: Remainder (1) is less than (12), so the form is correct.
Using Euclid’s Division Lemma, what will the remainder be when (b=1)?
Correct answer: A
Step 1: The remainder condition is (0 \le r < b). Step 2: When (b=1), we get (0 \le r < 1), so only (r=0) is possible. Step 3: Any integer divided by (1) leaves remainder (0).
QUIZ COMPLETE