Exercise: 2.5

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

**Answer:**

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

**(i) 6250**

Here, the unit’s digit = 0

Thus, the given number is divisible by 2.

**(ii) 984325 **

Here, the unit’s digit = 5

Thus, the given number is not divisible by 2.

**(iii) 367314**

Here, the unit’s digit = 4

Thus, the given number is divisible by 2.

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

**Answer:**

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

**(i) 70335**

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) 607439**

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) 9082746**

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.

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

**Answer:**

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

**(i) 7020 **

Here, the units 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) 56423 **

Here, the units 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) 732510**

Here, the units 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.

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

**Answer**:

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

**(i) 786532 **

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) 1020531**

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) 9801523**

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.

**5.) Test the divisibility of the following numbers by 8:**

**Answer:**

**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 digits 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 digits 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.

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

**Answer: **

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.

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

**Answer:**

**(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

Therefore, 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

Therefore, 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

Therefore, 1,00,00,001 is divisible by 11.

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

**Answer: **

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

**(i) 75*5 **

75*5 = 7515

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

**(ii) 35*64 **

35*64 = 35064

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

**(iii) 18 * 71**

18 * 71= 18171

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

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

**Answer:**

**(i) 67 *19**

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) 66784 * **

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) 538 * 8 **

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

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

**Answer:**

**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 x 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 [Because the missing number is a single digit]

Thus, missing number = 3

Hence, the smallest required number is 3.

**(ii) 467 x 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 [Because the missing number is a single digit]

Thus, missing number = 2

Hence, the smallest required number is 2.

**(iii) 9 x 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 [Because 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.

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

**Answer: **

(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.

**12.) Which of the following statement are true?**

**Answer:**

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

False. 12 is divisible by 3 but not by 9.

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

True.

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

False. 20 is divisible by 4 but not by 8.

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

True.

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

False. 12 is divisible by both 3 and 6 but it is not divisible by 18.

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

True.

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

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

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

True.

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

False. 4 and 9 are co-primes and both are composite numbers.

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

True.