GSEB Class 6 Maths Solutions Chapter 3 Playing With Numbers Exercise 3.3

Question 1. Using divisibility tests, determine which of the following numbers are divisible by 2; by 3; by 4; by 5; by 6; by 8; by 9; by 10; by 11 (say, yes or no):

Answer:

Exam Tip: To get full marks, memorize the divisibility rules for each number from 2 to 11 and apply them systematically. This helps ensure accuracy in determining divisibility.

Question 2. Using divisibility tests, determine which of the following numbers are divisible by 4; by 8:
(a) 572
(b) 726352
(c) 5500
(d) 6000
(e) 12159
(f) 14560
(g) 21084
(h) 31795072
(i) 1700
(j) 2150
Answer:
(a) 572
Divisibility by 4: The number formed by the last 2 digits is 72.
\( 4) 72 (18 \)
\( \quad \underline{-4} \)
\( \quad 32 \)
\( \quad \underline{-32} \)
\( \quad \quad 0 \)
\( \implies \) Remainder = 0. So, 72 is divisible by 4. Therefore, 572 is divisible by 4.
Divisibility by 8: The number formed by the last 3 digits is 572.
\( 8) 572 (71 \)
\( \quad \underline{-56} \)
\( \quad \quad 12 \)
\( \quad \quad \underline{-8} \)
\( \quad \quad \quad 4 \)
\( \implies \) Remainder = 4. So, 572 is not divisible by 8.

(b) 726352
Divisibility by 4: The number formed by the last two digits is 52.
\( 4) 52 (13 \)
\( \quad \underline{-4} \)
\( \quad 12 \)
\( \quad \underline{-12} \)
\( \quad \quad 0 \)
\( \implies \) Remainder = 0. So, 52 is divisible by 4. Therefore, 726352 is divisible by 4.
Divisibility by 8: The number formed by the last 3 digits is 352.
\( 8) 352 (44 \)
\( \quad \underline{-32} \)
\( \quad \quad 32 \)
\( \quad \quad \underline{-32} \)
\( \quad \quad \quad 0 \)
\( \implies \) Remainder = 0. So, 352 is divisible by 8. Therefore, 726352 is divisible by 8.

(c) 5500
Divisibility by 4: The last two digits are 0, 0.
\( \implies \) Thus, 5500 is divisible by 4.
Divisibility by 8: The number formed by the last three digits is 500.
\( 8) 500 (62 \)
\( \quad \underline{-48} \)
\( \quad \quad 20 \)
\( \quad \quad \underline{-16} \)
\( \quad \quad \quad 4 \)
\( \implies \) Remainder = 4. So, 500 is not divisible by 8. Therefore, 5500 is not divisible by 8.

(d) 6000
Divisibility by 4: Since the last two digits are 0, 0.
\( \implies \) Thus, 6000 is divisible by 4.
Divisibility by 8: Since the last three digits are 0, 0, 0.
\( \implies \) Thus, 6000 is divisible by 8.

(e) 12159
Divisibility by 4: The number formed by the last 2 digits is 59.
\( 4) 59 (14 \)
\( \quad \underline{-4} \)
\( \quad 19 \)
\( \quad \underline{-16} \)
\( \quad \quad 3 \)
\( \implies \) Remainder = 3. So, 59 is not divisible by 4. Therefore, 12159 is not divisible by 4.
Divisibility by 8: The number formed by the last 3 digits is 159.
\( 8) 159 (19 \)
\( \quad \underline{-8} \)
\( \quad 79 \)
\( \quad \underline{-72} \)
\( \quad \quad 7 \)
\( \implies \) Remainder = 7. So, 159 is not divisible by 8. Therefore, 12159 is not divisible by 8.

(f) 14560
Divisibility by 4: The number formed by the last two digits is 60.
\( 4) 60 (15 \)
\( \quad \underline{-4} \)
\( \quad 20 \)
\( \quad \underline{-20} \)
\( \quad \quad 0 \)
\( \implies \) Remainder = 0. So, 60 is divisible by 4. Therefore, 14560 is divisible by 4.
Divisibility by 8: The number formed by the last three digits is 560.
\( 8) 560 (70 \)
\( \quad \underline{-56} \)
\( \quad \quad 00 \)
\( \quad \quad \underline{-00} \)
\( \quad \quad \quad 0 \)
\( \implies \) Remainder = 0. So, 560 is divisible by 8. Therefore, 14560 is divisible by 8.

(g) 21084
Divisibility by 4: The number formed by the last two digits is 84.
\( 4) 84 (21 \)
\( \quad \underline{-8} \)
\( \quad \quad 04 \)
\( \quad \quad \underline{-04} \)
\( \quad \quad \quad 0 \)
\( \implies \) Remainder = 0. So, 84 is divisible by 4. Therefore, 21084 is divisible by 4.
Divisibility by 8: The number formed by the last three digits is 084 = 84.
\( 8) 84 (10 \)
\( \quad \underline{-8} \)
\( \quad \quad 04 \)
\( \quad \quad \underline{-00} \)
\( \quad \quad \quad 4 \)
\( \implies \) Remainder = 4. So, 84 is not divisible by 8. Therefore, 21084 is not divisible by 8.

(h) 31795072
Divisibility by 4: The number formed by the last two digits is 72.
\( 4) 72 (18 \)
\( \quad \underline{-4} \)
\( \quad 32 \)
\( \quad \underline{-32} \)
\( \quad \quad 0 \)
\( \implies \) Remainder = 0. So, 72 is divisible by 4. Therefore, 31795072 is divisible by 4.
Divisibility by 8: The number formed by the last three digits is 072 = 72.
\( 8) 72 (9 \)
\( \quad \underline{-72} \)
\( \quad \quad 0 \)
\( \implies \) Remainder = 0. So, 72 is divisible by 8. Therefore, 31795072 is divisible by 8.

(i) 1700
Divisibility by 4: The number formed by the last two digits is 00.
\( \implies \) Thus, 1700 is divisible by 4.
Divisibility by 8: The number formed by the last three digits is 700.
\( 8) 700 (87 \)
\( \quad \underline{-64} \)
\( \quad \quad 60 \)
\( \quad \quad \underline{-56} \)
\( \quad \quad \quad 4 \)
\( \implies \) Remainder = 4. So, 700 is not divisible by 8. Therefore, 1700 is not divisible by 8.

(j) 2150
Divisibility by 4: The number formed by the last two digits is 50.
\( 4) 50 (12 \)
\( \quad \underline{-4} \)
\( \quad 10 \)
\( \quad \underline{-8} \)
\( \quad \quad 2 \)
\( \implies \) Remainder = 2. So, 50 is not divisible by 4. Therefore, 2150 is not divisible by 4.
Divisibility by 8: The number formed by the last three digits is 150.
\( 8) 150 (18 \)
\( \quad \underline{-8} \)
\( \quad 70 \)
\( \quad \underline{-64} \)
\( \quad \quad 6 \)
\( \implies \) Remainder = 6. So, 150 is not divisible by 8. Therefore, 2150 is not divisible by 8.

Exam Tip: For divisibility by 4, check the last two digits. For divisibility by 8, check the last three digits. This is a quick way to determine if larger numbers can be divided evenly.

Question 3. Using divisibility tests, determine which of following numbers are divisible by 6:
(a) 297144
(b) 1258
(c) 4335
(d) 61233
(e) 901352
(f) 438750
(g) 1790184
(h) 12583
(i) 639210
(j) 17852
Answer:
(a) 297144
Divisibility by 2: The unit's digit is 4, which is divisible by 2. Thus, 297144 is divisible by 2.
Divisibility by 3: The sum of the digits is \( 2 + 9 + 7 + 1 + 4 + 4 = 27 \), which is divisible by 3. Thus, 297144 is divisible by 3.
Conclusion: Since 297144 is divisible by both 2 and 3, it is divisible by 6.

(b) 1258
Divisibility by 2: The unit's digit is 8, which is an even number. Thus, 1258 is divisible by 2.
Divisibility by 3: The sum of the digits is \( 1 + 2 + 5 + 8 = 16 \), which is not divisible by 3.
Conclusion: Since 1258 is divisible by 2 but not by 3, it is not divisible by 6.

(c) 4335
Divisibility by 2: The unit's digit is 5, which is not divisible by 2. Thus, 4335 is not divisible by 2.
Divisibility by 3: The sum of the digits is \( 4 + 3 + 3 + 5 = 15 \), which is divisible by 3. Thus, 4335 is divisible by 3.
Conclusion: Since 4335 is divisible by 3 but not by 2, it is not divisible by 6.

(d) 61233
Divisibility by 2: The unit's digit is 3, which is not divisible by 2. Thus, 61233 is not divisible by 2.
Divisibility by 3: The sum of the digits of 61233 is \( 6 + 1 + 2 + 3 + 3 = 15 \), which is divisible by 3. Thus, 61233 is divisible by 3.
Conclusion: Since 61233 is divisible by 3 but not by 2, it is not divisible by 6.

(e) 901352
Divisibility by 2: The unit's digit is 2, which is divisible by 2. Thus, 901352 is divisible by 2.
Divisibility by 3: The sum of digits of 901352 is \( 9 + 0 + 1 + 3 + 5 + 2 = 20 \), which is not divisible by 3. Thus, 901352 is not divisible by 3.
Conclusion: Since 901352 is divisible by 2 but not by 3, it is not divisible by 6.

(f) 438750
Divisibility by 2: The unit's digit is 0. Thus, 438750 is divisible by 2.
Divisibility by 3: The sum of digits of 438750 is \( 4 + 3 + 8 + 7 + 5 + 0 = 27 \), which is divisible by 3. Thus, 438750 is divisible by 3.
Conclusion: Since 438750 is divisible by both 2 and 3, it is divisible by 6.

(g) 1790184
Divisibility by 2: The unit's digit is 4, which is divisible by 2. Thus, 1790184 is divisible by 2.
Divisibility by 3: The sum of digits of 1790184 is \( 1 + 7 + 9 + 0 + 1 + 8 + 4 = 30 \), which is divisible by 3. Thus, 1790184 is divisible by 3.
Conclusion: Since 1790184 is divisible by 2 and 3 both, it is divisible by 6.

(h) 12583
Divisibility by 2: The unit's digit is 3, which is not divisible by 2. Thus, 12583 is not divisible by 2.
Divisibility by 3: The sum of digits of 12583 is \( 1 + 2 + 5 + 8 + 3 = 19 \), which is not divisible by 3.
Conclusion: Since 12583 is neither divisible by 2 nor by 3, it is not divisible by 6.

(i) 639210
Divisibility by 2: The unit's digit is 0. Thus, 639210 is divisible by 2.
Divisibility by 3: The sum of the digits of 639210 is \( 6 + 3 + 9 + 2 + 1 + 0 = 21 \), which is divisible by 3.
Conclusion: Since 639210 is divisible by both 2 and 3, it is divisible by 6.

(j) 17852
Divisibility by 2: The unit's digit is 2, which is divisible by 2. Thus, 17852 is divisible by 2.
Divisibility by 3: The sum of digits of 17852 is \( 1 + 7 + 8 + 5 + 2 = 23 \), which is not divisible by 3. Thus, 17852 is not divisible by 3.
Conclusion: Since 17852 is divisible by 2 but not by 3, it is not divisible by 6.

Exam Tip: To check divisibility by 6, remember that a number must be divisible by both 2 and 3. Always test both conditions carefully.

Question 4. Using divisibility tests, determine which of the following numbers are divisible by 11.
(a) 5445
(b) 10824
(c) 7138965
(d) 70169308
(e) 10000001
(f) 901153
Answer:
(a) 5445
Sum of digits at odd places from the right = \( 5 + 4 = 9 \).
Sum of digits at even places from the right = \( 4 + 5 = 9 \).
Difference of these two sums = \( 9 - 9 = 0 \).
Since the difference is 0, 5445 is divisible by 11.

(b) 10824
Sum of the digits at odd places from the right = \( 4 + 8 + 1 = 13 \).
Sum of digits at even places from the right = \( 2 + 0 = 2 \).
Difference of these two sums = \( 13 - 2 = 11 \).
Since the difference is a multiple of 11, 10824 is divisible by 11.

(c) 7138965
Sum of digits at odd places from the right = \( 5 + 9 + 3 + 7 = 24 \).
Sum of digits at even places from the right = \( 6 + 8 + 1 = 15 \).
Difference of these two sums = \( 24 - 15 = 9 \).
Since 9 is not a multiple of 11, 7138965 is not divisible by 11.

(d) 70169308
From the right, Sum of the digits at odd places = \( 8 + 3 + 6 + 0 = 17 \).
Sum of the digits at even places = \( 0 + 9 + 1 + 7 = 17 \).
Difference of these two sums = \( 17 - 17 = 0 \).
Since the difference is 0, 70169308 is divisible by 11.

(e) 10000001
From the right, Sum of the digits at odd places = \( 1 + 0 + 0 + 0 = 1 \).
Sum of the digits at even places = \( 0 + 0 + 0 = 1 \).
The difference of these two sums = \( 1 - 1 = 0 \).
Since the difference is 0, 10000001 is divisible by 11.

(f) 901153
From the right, The sum of digits at odd places = \( 3 + 1 + 0 = 4 \).
The sum of digits at even places = \( 5 + 1 + 9 = 15 \).
Difference of these two sums = \( 15 - 4 = 11 \).
Since the difference is a multiple of 11, 901153 is divisible by 11.

Exam Tip: The divisibility test for 11 involves checking the difference between the sum of digits at odd places and the sum of digits at even places. If the difference is 0 or a multiple of 11, the number is divisible by 11.

Question 5. Write the smallest digit and the greatest digit in the blank space of each of the following numbers so that the number formed is divisible by 3:
(a) ....... 6724
(b) 4765 ...... 2
Answer:
(a) \( \_6724 \)
Sum of the given digits = \( 6 + 7 + 2 + 4 = 19 \).
Multiples of 3 greater than 19 are 21, 24, 27, and 30.
To find the smallest digit, calculate: \( 21 - 19 = 2 \).
To find the next possible digit: \( 24 - 19 = 5 \).
To find the next possible digit: \( 27 - 19 = 8 \).
To find the next possible digit: \( 30 - 19 = 11 \) (which is not a single digit).
The smallest required digit is 2.
The greatest required digit is 8.

(b) \( 4765\_2 \)
Sum of the given digits = \( 4 + 7 + 6 + 5 + 2 = 24 \), which is a multiple of 3.
The smallest required digit is 0 (because \( 24 + 0 = 24 \), which is divisible by 3).
Next multiples of 3 are 27, 30, 33, 36.
To find the next possible digit, calculate: \( 27 - 24 = 3 \).
To find the next possible digit: \( 30 - 24 = 6 \).
To find the next possible digit: \( 33 - 24 = 9 \).
To find the next possible digit: \( 36 - 24 = 12 \) (which is not a single digit).
The largest required digit is 9.

Exam Tip: For divisibility by 3, remember that the sum of the digits must be divisible by 3. Find multiples of 3 that are slightly greater than the sum of known digits to determine the missing digit.

Question 6. Write a digit in the blank space of each of the following numbers so that the number formed is divisible by 11:
(a) \( 92\_389 \)
(b) \( 8\_9484 \)
Answer:
(a) \( 92\_389 \)
We have:
Sum of the digits at odd places (from the right) = \( 9 + 3 + 2 = 14 \).
Sum of digits at even places (from the right) = \( 8 + [\text{Missing digit}] + 9 = [\text{Missing digit}] + 17 \).
Difference of these sums = \( [\text{Missing digit}] + 17 - 14 = [\text{Missing digit}] + 3 \).
For the given number to be divisible by 11, \( [\text{Missing digit}] + 3 \) must be a multiple of 11.
The smallest non-negative multiple of 11 that is \( \geq 3 \) is 11 itself.
So, Missing digit \( + 3 = 11 \)
\( \implies \) Missing digit \( = 11 - 3 = 8 \).
Thus, the number is 928389.

(b) \( 8\_9484 \)
Sum of the digits at odd places (from the right) = \( 4 + 4 + [\text{Missing digit}] = [\text{Missing digit}] + 8 \).
Sum of the digits at even places (from the right) = \( 8 + 9 + 8 = 25 \).
Difference of these sums = \( 25 - ([\text{Missing digit}] + 8) \).
\( = 25 - [\text{Missing digit}] - 8 \).
\( = 17 - [\text{Missing digit}] \).
For the given number to be divisible by 11, \( 17 - [\text{Missing digit}] \) must be a multiple of 11.
Possible multiples of 11 are 0, 11, -11, etc.
If \( 17 - [\text{Missing digit}] = 0 \), then Missing digit \( = 17 \) (not a single digit).
If \( 17 - [\text{Missing digit}] = 11 \), then Missing digit \( = 17 - 11 = 6 \).
If \( 17 - [\text{Missing digit}] = -11 \), then Missing digit \( = 17 + 11 = 28 \) (not a single digit).
Thus, the missing digit is 6.
The given number is 869484.

Exam Tip: When using the divisibility test for 11 with a missing digit, remember to consider both cases where the difference could be 0, 11, -11, or other multiples, and select the single digit that fits the number.