Class 6 Playing with Numbers Exercise 2.1-2

\begin{array}{l} \text {Q6. Without actual division show that 11 is a factor of each of the following numbers: } \\ (i) \quad 1111 \quad (ii) \quad 11011 \\ (iii) \quad 110011 \quad (iv) \quad 1100011 \\ \\\text {Sol. (i) } \quad 1111= 1100 + 11 \\ = 11 \times 100 + 11 \times \\ = 11 \times (100 + 1) \\ = 11 \times 101 \\ \text {Clearly, we can see 11 is factor of } 11 \times 101 \\ \text {Hence, 11 is a factor of 1111} \\ \\\text {(ii) } \quad 11011= 11000 + 11 \\ = 11 \times 1000 + 11 \times \\ = 11 \times (1000 + 1) \\ = 11 \times 1001 \\ \text {Clearly, we can see 11 is factor of } 11 \times 1001 \\ \text {Hence, 11 is a factor of 11011} \\ \\\text {(iii) } \quad 110011= 110000 + 11 \\ = 11 \times 10000 + 11 \times \\ = 11 \times (10000 + 1) \\ = 11 \times 10001 \\ \text {Clearly, we can see 11 is factor of } 11 \times 10001 \\ \text {Hence, 11 is a factor of 110011} \\ \\\text {(iv) } \quad 1100011= 1100000 + 11 \\ = 11 \times 100000 + 11 \times \\ = 11 \times (100000 + 1) \\ = 11 \times 100001 \\ \text {Clearly, we can see 11 is factor of } 11 \times 100001 \\ \text {Hence, 11 is a factor of 1100011} \\ \\\text {Q7. Without actual division show that each of the following numbers is divisible by 5 : } \\ (i) \quad 55 \quad (ii) \quad 555 \\ (iii) \quad 5555 \quad (iv) \quad 50005 \\ \\\text {Sol. A number will be divisible by 5 if the unit’s digit } \\ \text { of that number is either 0 or 5.} \\ \\\text {(i) In 55, the unit’s digit is 5. } \\ \text {Hence, 55 is divisible by 5.} \\ \\\text {(ii) In 555, the unit’s digit is 5. } \\ \text {Hence, 5555 is divisible by 5.} \\ \\ \text {(iii) In 555, the unit’s digit is 5. } \\ \text {Hence, 5555 is divisible by 5.} \\ \\\text {(iv) In 50005, the unit’s digit is 5. } \\ \text {Hence, 50005 is divisible by 5.} \\ \\\text {Q8. Is there any natural number having no factor at all? } \\\text {Sol. No, because all natural numbers are a factor of itself.} \\ \\\text {Q9. Find numbers between 1 and 100 having exactly three factors. } \\ \\\text {Sol. The numbers between 1 and 100 having exactly three factors are 4,9,25 and 49} \\ \text {The factors of 4 are 1,2 and 4} \\ \text {The factors of 9 are 1,3 and 9} \\ \text {The factors of 25 are 1,5 and 25} \\ \text {The factors of 49 are 1,7 and 49} \\ \\\text {Q10. Sort out even and odd numbers: } \\ (i) \quad 42 \quad (ii) \quad 89 \\ (iii) \quad 144 \quad (iv) \quad 321 \\ \\\text {Sol. A number which is exactly divisible by 2 is called an even number.} \\ \therefore \text { 42 and 144 are even numbers.} \\\text {A number which is not exactly divisible by 2 is called an odd number.} \\ \therefore \text {89 and 321 are odd numbers.} \\ \end{array}