**Playing with Numbers**

**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 4; by 11 (say yes or no) :**

**Solution:**

**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

**Solution:**

We know that a number is divisible by 4, if the number formed by its digits in ten’s and unit’s place is divisible by 4.**(a)** In 572, 72 is divisible by 4. So, 572 is divisible by 4.**(b)** In 726352, 52 is divisible by 4. So, it is divisible by 4.**(c)** In 5500, 00 is divisible by 4. So, it is divisible by 4.**(d)** In 6000, 00 is divisible by 4. So, it is divisible by 4.**(e)** In 12159, 59 is not divisible by 4. So, it is not divisible by 4.**(f)** In 14560, 60 is divisible by 4. So, it is divisible by 4.**(g)** In 21084, 84 is divisible by 4. So, it is divisible by 4.**(h)** In 31795072, 72 is divisible by 4. So, it is divisible by 4.**(i)** In 1700,00 is divisible by 4. So, it is divisible by 4.**(j)** In 2150, 50 is not divisible by 4. So, it is not divisible by 4.

Also, we know that a number is divisible by 8, if the number formed by its hundred’s, ten’s and unit’s places is divisible by 8.

**(a)** 572 is not divisible by 8.**(b)** In 726352, 352 is divisible by 8. So, it divisible by 8.**(c)** In 5500, 500 is not divisible by 8. So, it is not divisible by 8.**(d)** In 6000, 000 is divisible by 8. So, it is divisible by 8.**(e)** In 12159, 159 is not divisible by 8. So, it is not divisible by 8.**(f)** In 14560, 560 is divisible by 8. So, it is divisible by 8.**(g)** In 21084, 084 is not divisible by 8. So, it is not divisible by 8.**(h)** In 31795072, 072 is divisible by 8. So, it is divisible by 8.**(i)** In 1700, 700 is not divisible by 8. So, it is not divisible by 8.**(j)** In 2150, 150 is not divisible by 8. So, it is not divisible by 8.

**Question 3.****Using divisibility tests, determine which of the followingnumbers are divisible by 6 : .**

**(a)**297144

**(b)**1258

**(c)**4335

**(d)**61233

**(e)**901352

**(f)**438750

**(g)**1790184

**(h)**12583

**(i)**639210

**(j)**17852

**Solution:**

We know that a number is divisible by 6, if it is divisible by 2 and 3 both.**(a)** Given number = 297144

Its unit’s digit is 4. So, it is divisible by 2.

Sum of its digits = 2+ 9+ 7 + 1 + 4 + 4 = 27, which is divisible by 3.

∴ 297144 is divisible by 6.

**(b)** Given number =1258

Its unit’s digit is 8. So, it is divisible by 2.

Sum of its digits = 1+ 2 + 5 + 8 = 16, which is not divisible by 3.

∴ 1258 is not divisible by 6.

**(c)** Given number = 4335 .

Its unit’s digit is 5. So, it is not divisible by 2.

∴ 4335 is also not divisible by 6.

**(d)** Given number = 61233

Its unit’s digit is 3. So, it is not divisible by 2.

∴ 61233 is also not divisible by 6.

**(e)** Given number = 901352

Its unit’s digit is 2. So, it is divisible by 2.

Sum of its digits = 9+ 0 + 1 + 345 + 2 = 20, which is not divisible by 3.

∴ 901352 is not divisible by 6.

**(f)** Given number = 438750

Its unit’s digit is 0. So, it is divisible by 2.

Sum of its digits = 4+ 3 + 8 + 7 + 5 + 0=27, which is divisible by 3.

∴ 438750 is divisible by 6.

**(g)** Given number = 1790184

Its unit’s digit is 4. So, it is divisible by 2.

Sum of its digits = 1+ 7 + 9+ 0+ 1+ 8 + 4 = 30, which is divisible by 3.

∴ 1790184 is divisible by 6.

**(h)** Given number = 12583 .

Its unit’s digit is 3. So, it is not divisible by 2.

∴ 12583 is not divisible by 6.

**(i)** Given number = 639210

Its unit’s digit is 0. So, it is divisible by 2.

Sum of its digits = 6+ 3+ 9+ 2 + 1 + 0 = 21, which is divisible by 3.

∴ 639210 is divisible by 6.

**(j)** Given number = 17852

Its unit’s digit is 2. So, it is divisible by 2.

Sum of its digits = 1 + 7 + 8 + 5 + 2 = 23, which is not divisible by 3.

∴ 17852 is not divisible by 6.

**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

**Solution:**

We know that a number is divisible by 11, if the difference in odd places (from the right) and the sum of its digits in even places (from the right) is either 0 or a multiple of 11.**(a)** Given number = 5445

Sum of its digits at odd places = 5 + 4 = 9

Sum of its digit at even places = 4 + 5 = 9

Difference of these two sums = 9 – 9 = 0

∴ 5445 is divisible by 11.

**(b)** Given number = 10824

Sum of its digits at odd places = 4+ 8 + 1 = 13

Sum of its digits at even places =2 + 0 =2

Difference of these two sums =13 – 2 = 11, which is a multiple of 11.

∴ 10824 is divisible by 11.

**(c)** Given number = 7138965

Sum of its digits at odd places = 5+ 9+ 3 + 7= 24

Sum of its digits at even places = 6+ 8 + 1 = 15

Difference of these two sums = 24 – 15 = 9, which is not a multiple of 11.

∴ 7138965 is not divisible by 11.

**(d)** Given number = 70169308

Sum of its digits at odd places = 8 + 3 + 6 + 0=17

Sum of its digits at even places = 0 + 9 + 1 + 7 = 17

Difference of these two sums =17 – 17 = 0

∴ 70169308 is divisible by 11.

**(e)** Given number = 10000001

Sum of its digits at odd places = 1 + 0 + 0 + 0 = 1

Sum of its digits at even places = 0 + 0 + 0 + 1 = 1

Difference of these two sums = 1 – 1 = 0

∴ 10000001 is divisible by 11.

**(f)** Given number = 901153

Sum of its digits at odd places = 3 + 1 + 0 = 4

Sum of its digits at even places = 5 + 1 + 9=15

Difference of these two sums =15 – 4 = 11,

which is a multiple of 11.

∴ 901153 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

**Solution:**

We know that a number divisible by 3, if the sum of its digits is divisible by 3.**(a) … 6724**

For … 6724, we have 6 + 7 + 2 + 4 =19, we add 2 to 19, the resulting number 21 will be divisible by 3.

∴ The required smallest digit is 2.

Again, if we add 8 to 19, the resulting number 27 will be divisible by 3. .’. The required largest digit is 8.

**(b) 4765 … 2**

For 4765 … 2, we have 4 + 7 + 6 + 5 + 2 = 24, it is divisible by 3.

Hence the required smallest digit is 0.

Again, if we add 9 to 24, the resulting number 33 will be divisible by 3.

∴ The required largest digit is 9.

**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

**Solution:**

We know that a number is divisible by 11, if the difference of the sum of its digits at odd places and the sum of its digits at even places is either 0 or divisible by 11.**(a) 92 … 389**

For 92 … 389, sum of the digits at odd places and sum of digits at even places

= 9 + 3 + 2 = 14

= 8 + required digit + 9

= required digit+ 17

Difference between these sums

= required digit + 17 – 14

= required digit + 3

For (required digit + 3) to become 11, we must have the required digit as 8 (∵ 3+ 8 gives 11).

Hence, the required smallest digit = 8 .

**(b) 8…9484**

For 8 … 9484, sum of the digits at odd places

= 4 + 4 + required digit

= 8 + required digit

and sum of digits at even places

= 8 + 9 + 8 = 25

Difference between these sums

= 25 – (8 + required digit)

= 17- required digit

For (17 — required digit) to become 11 we must have the required digit as 6 (∵ 17-6 =11).

Hence the required smallest digit = 6