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**Q.1. ****Test the divisibility of the following numbers by 2:**

**(i) 6520**

**(ii) 984325**

**(iii) 367314**

**Ans: **Rule: A natural number is divisible by 2 if its unit digit is 0, 2, 4, 6, or 8.

**(i)** Here, the unit's digit = 0

Thus, the given number is divisible by 2.

**(ii) **Here, the unit's digit = 5

Thus, the given number is not divisible by 2.

**(iii) **Here, the unit's digit = 4

Thus, the given number is divisible by 2.

**Q.2. ****Test the divisibility of the following numbers by 3:**

**(i) 70335**

**(ii) 607439**

**(iii) 9082746**

**Ans: ****Rule: **A number is divisible by 3 if the sum of its digits is divisible by 3.

**(i) **Here, the sum of the digits in the given number = 7 + 0 + 3 + 3 + 5 = 18 which is divisible by 3.

Thus, 70,335 is divisible by 3.

**(ii) **Here, the sum of the digits in the given number = 6 + 0 + 7 + 4 + 3 + 9 = 29 which is not divisible by 3.

Thus, 6,07,439 is not divisible by 3.

**(iii) **Here, the sum of the digits in the given number = 9 + 0 + 8 + 2 + 7 + 4 + 6 = 36 which is divisible by 3.

Thus, 90,82,746 is divisible by 3.

**Q.3. ****Test the divisibility of the following numbers by 6:**

**(i) 7020**

**(ii) 56423**

**(iii) 732510**

**Ans: ****Rule: **A number is divisible by 6 if it is divisible by 2 as well as 3.

**(i)** Here, the unit's digit = 0

Thus, the given number is divisible by 2.

Also, the sum of the digits = 7 + 0 + 2 + 0 = 9 which is divisible by 3. So, the given number is divisible by 3.

Hence, 7,020 is divisible by 6.

**(ii)** Here, the unit's digit = 3

Thus, the given number is not divisible by 2.

Also, the sum of the digits = 5 + 6 + 4 + 2 + 3 = 20 which is not divisible by 3. So, the given number is not divisible by 3.

Since 3,56,423 is neither divisible by 2 nor by 3, it is not divisible by 6.

**(iii) ** Here, the unit's digit = 0

Thus, the given number is divisible by 2.

Also, the sum of the digits = 7 + 3 + 2 + 5 + 1 + 0 = 18 which is divisible by 3. So, the given number is divisible by 3.

Hence, 7,32,510 is divisible by 6.

**Q.4. ****Test the divisibility of the following numbers by 4:**

**(i) 786532**

**(ii) 1020531**

**(iii) 9801523**

**Ans: ****Rule:** A natural number is divisible by 4 if the number formed by its last two digits is divisible by 4.

**(i) **Here, the number formed by the last two digits is 32 which is divisible by 4.

Thus, 7,86,532 is divisible by 4.

**(ii) **Here, the number formed by the last two digits is 31 which is not divisible by 4.

Thus, 10,20,531 is not divisible by 4.

**(iii)** Here, the number formed by the last two digits is 23 which is not divisible by 4.

Thus, 98,01,523 is not divisible by 4.

**Q.5. ****Test the divisibility of the following numbers 8:**

**(i) 8364**

**(ii) 7314**

**(iii) 36712**

**Ans: ****Rule:** A number is divisible by 8 if the number formed by its last three digits is divisible by 8.

**(i) **The given number = 8364

The number formed by its last three digit is 364 which is not divisible by 8.

Therefore, 8,364 is not divisible by 8.

**(ii) **The given number = 7314

The number formed by its last three digit is 314 which is not divisible by 8.

Therefore, 7,314 is not divisible by 8.

**(iii)** The given number = 36712

Since the number formed by its last three digit = 712 which is divisible by 8.

Therefore, 36,712 is divisible by 8.

**Q.6. ****Test the divisibility of the following numbers by 9:**

**(i) 187245**

**(ii) 3478**

**(iii) 547218**

**Ans: ****Rule: **A number is divisible by 9 if the sum of its digits is divisible by 9.

**(i)** The given number = 187245

The sum of the digits in the given number = 1 + 8 + 7 + 2 + 4 + 5 = 27 which is divisible by 9.

Therefore, 1,87,245 is divisible by 9.

**(ii)** The given number = 3478

The sum of the digits in the given number = 3 + 4 + 7 + 8 = 22 which is not divisible by 9.

Therefore, 3,478 is not divisible by 9.

**(iii) **The given number = 547218

The sum of the digits in the given number = 5 + 4 + 7 + 2 + 1 + 8 = 27 which is divisible by 9.

Therefore, 5,47,218 is divisible by 9.

**Q.7. ****Test the divisibility of the following numbers by 11:**

**(i) 5335**

**(ii) 70169803**

**(iii) 10000001**

**Ans: **

**(i) **The given number is 5,335.

The sum of the digit at the odd places = 5 + 3 = 8

The sum of the digits at the even places = 3 + 5 = 8

Their difference = 8 − 8 = 0

∴ 5,335 is divisible by 11.

**(ii) **The given number is 7,01,69,803.

The sum of the digit at the odd places = 7 + 1 + 9 + 0 = 17

The sum of the digits at the even places = 0 + 6 + 8 + 3 = 17

Their difference = 17 − 17 = 0

∴ 7,01,69,803 is divisible by 11.

**(iii) **The given number is 1,00,00,001.

The sum of the digit at the odd places = 1 + 0 + 0 + 0 = 1

The sum of the digits at the even places = 0 + 0 + 0 + 1 = 1

Their difference = 1 − 1 = 0

∴ 1,00,00,001 is divisible by 11.

**Q.8. ****In each of the following numbers, replace * by the smallest number to make it divisible by 3:**

**(i) 75 * 5**

**(ii) 35 * 64**

**(iii) 18 * 71**

**Ans: **We can replace the * by the smallest number to make the given numbers divisible by 3 as follows:

**(i) **75*5 = 7515

As 7 + 5 + 1 + 5 = 18, it is divisible by 3.

**(ii) **35*64 = 35064

As 3 + 5 +6 + 4 = 18, it is divisible by 3.

**(iii) **18*71 = 18171

As 1 + 8 + 1 + 7 + 1 = 18, it is divisible by 3.

**Q.9.****In each of the following numbers, replace * by the smallest number to make it divisible by 9:**

**(i) 67 * 19**

**(ii) 66784 ***

**(iii) 538 * 8**

**Ans: **

**(i)** Sum of the given digits = 6 + 7 + 1 + 9 = 23

The multiple of 9 which is greater than 23 is 27.

Therefore, the smallest required number = 27 − 23 = 4

**(ii)** Sum of the given digits = 6 + 6 + 7 + 8 + 4 = 31

The multiple of 9 which is greater than 31 is 36.

Therefore, the smallest required number = 36 − 31 = 5

**(iii)** Sum of the given digits = 5 + 3 + 8 + 8 = 24

The multiple of 9 which is greater than 24 is 27.

Therefore, the smallest required number = 27 − 24 = 3

**Q.10.**** In each of the following numbers. replace * by the smallest number to make it divisible by 11:**

**(i) 86 * 72**

**(ii) 467 * 91**

**(iii) 9 * 8071**

**Ans: ****Rule:** A number is divisible by 11 if the difference of the sums of the alternate digits is either 0 or a multiple of 11.

**(i)** 86 × 72

Sum of the digits at the odd places = 8 + missing number + 2 = missing number + 10

Sum of the digits at the even places = 6 + 7 = 13

Difference = [missing number + 10 ] − 13 = Missing number − 3

According to the rule, missing number − 3 = 0 [∵ the missing number is a single digit]

Thus, missing number = 3

Hence, the smallest required number is 3.

**(ii)** 467 × 91

Sum of the digits at the odd places = 4 + 7 + 9 = 20

Sum of the digits at the even places = 6 + missing number + 1 = missing number + 7

Difference = 20 − [missing number + 7] = 13 − missing number

According to rule, 13 − missing number = 11 [∵ the missing number is a single digit]

Thus, missing number = 2

Hence, the smallest required number is 2.

**(iii) **9 × 8071

Sum of the digits at the odd places = 9 + 8 + 7 = 24

Sum of the digits at the even places = missing number + 0 + 1 = missing number + 1

Difference = 24 − [missing number + 1] = 23 − missing number

According to rule, 23 − missing number = 22 [∵ 22 is a multiple of 11 and the missing number is a single digit]

Thus, missing number = 1

Hence, the smallest required number is 1.

**Q.11. ****Given an example of a number which is divisible by**

**(i) 2 but not by 4.**

**(ii) 3 but not by 6.**

**(iii) 4 but not by 8.**

**(iv) both 4 and 8 but not by 32.**

** ****Ans: **

**(i) **A number which is divisible by 2 but not by 4 is 6.

**(ii) **A number which is divisible by 3 but not by 6 is 9.

**(iii) **A number which is divisible by 4 but not by 8 is 28.

**(iv) **A number which is divisible by 4 and 8 but not by 32 is 48.

**Q.12. ****Which of the following statements are true?**

**(i) If a number is divisible by 3, it must be divisible by 9.**

**(ii) If a number is divisible by 9, it must be divisible by 3.**

**(iii) If a number is divisible by 4, it must be divisible by 8.**

**(iv) If a number is divisible by 8, it must be divisible by 4.**

**(v) A number is divisible by 18, if it is divisible by both 3 and 6.**

**(vi) If a number is divisible by both 9 and 10, it must be divisible by 90.**

**(vii) If a number exactly divides the sum of two numbers, it must exactly divide the numbers separately.**

**(viii) If a number divides three numbers exactly, it must divide their sum exactly.**

**(ix) If two numbers are co-prime, at least one of them must be a prime number**

**(x) The sum of two consecutive odd numbers is always divisible by 4.**

**Ans: **

**(i) **False. 12 is divisible by 3 but not by 9.

**(ii) **True.

**(iii) **False. 20 is divisible by 4 but not by 8.

**(iv)** True.

**(v)** False. 12 is divisible by both 3 and 6 but it is not divisible by 18.

**(vi) **True.

**(vii)** False. 10 divides the sum of 18 and 2 (i.e., 20) but 10 divides neither 18 nor 2.

**(viii)** True.

**(ix) **False. 4 and 9 are co-primes and both are composite numbers.

**(x) **True.

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