Class 6 Playing with Numbers Exercise 2.1-1

\begin{array}{l}\text {Q1. Define: } \\ \text {(i) factor } \\ \text {(ii) multiple Give four examples of each. } \\ \\\text {Sol. (i) A factor of a number is an exact divisor of that number. } \\\text {2 and 3 are factors of 6 i.e. } 2 \times 3 =6 \\ \text {2 and 4 are factors of 8 i.e. } 2 \times 4 =8 \\ \text {2 and 5 are factors of 10 i.e. } 2 \times 5=10 \\ \text {2 and 6 are factors of 12 i.e. } 2 \times 6=12 \\\text {(ii) A multiple of a number is a number obtained by multiplying it by a natural number.} \\ \text {10 is a multiple of 5 i.e. } 2 \times 5=10 \\ \text {12 is a multiple of 3 i.e. } 4 \times 3=12 \\ \text {8 is a multiple of 4 i.e. } 2 \times 4=8 \\ \text {21 is a multiple of 7 i.e. } 3 \times 7=21 \\ \\\text {Q2. Write all factors of each of the following numbers: } \\ (i) \quad \quad 60 \quad (ii) \quad \quad 76 \quad (iii) \quad \quad 125 \quad (iv) \quad \quad 729 \\ \\\text {Sol. (i) 60 can be written as } \\ 1 \times 60=60 \\ 2 \times 30=60 \\ 3 \times 20=60 \\ 4 \times 15=60 \\ 5 \times 12=60 \\ 6 \times 10=60 \\\therefore \text {The factors of 60 are 1,2,3,4,5,6,10,12,15,20,30 and 60} \\ \\\text {(ii) 76 It can be written as } \\ 1 \times 76=76 \\ 2 \times 38=76 \\ 4 \times 19=76 \\ \therefore \text {The factors of 76 are 1,2,4,19,38 and 76} \\ \\\text {(iii) 125 It can be written as } \\ 1 \times 125=125 \\ 5 \times 25=125 \\ therefore, \text {The factors of 125 are 1,5,25 and 125} \\ \\\text {(iv) 729 It can be written as } \\ 1 \times 729=729 \\ 3 \times 243=729 \\ 9 \times 81=729 \\ 27 \times 27=729 \\ \therefore \text {The factors of 729 are 1,3,9,27,81,243 and 729 } \\ \\\text {Q3. Write first five multiples of each of the following numbers: } \\ (i) \quad 25 \quad (ii) \quad 35 \quad (iii) \quad 45 \quad (iv) \quad 40 \\ \\\text {Sol. (i) In order to obtain the first five multiples of 25, } \\ \text {we multiply 25 by 1,2,3,4 and 5 respectively} \\ 1 \times 25= 25 \\ 2 \times 25=50 \\ 3 \times 25=75 \\ 4 \times 25=100 \\ 5 \times 25=125 \\\text {Hence, first five multiples of 25 are 25, 50, 75, 100 and 125 respectively.} \\ \\\text {(ii) In order to obtain the first five multiples of 35, } \\ \text {we multiply 35 by 1,2,3,4 and 5 respectively} \\ 1 \times 35= 35 \\ 2 \times 35=70 \\ 3 \times 35=105 \\ 4 \times 35=140 \\ 5 \times 35=175 \\\text {Hence, first five multiples of 35 are 35, 70, 105, 140 and 175 respectively.} \\ \\\text {(iii) In order to obtain the first five multiples of 45, } \\ \text {we multiply 45 by 1,2,3,4 and 5 respectively} \\ 1 \times 45= 45 \\ 2 \times 45=90 \\ 3 \times 45=135 \\ 4 \times 45=180 \\ 5 \times 45=225 \\\text {Hence, first five multiples of 45 are 45, 90, 135, 180 and 225 respectively.} \\ \\\text {(iv) In order to obtain the first five multiples of 40, } \\ \text {we multiply 40 by 1,2,3,4 and 5 respectively} \\ 1 \times 40= 40 \\ 2 \times 40=80 \\ 3 \times 40=120 \\ 4 \times 40=160 \\ 5 \times 40=200 \\\text {Hence, first five multiples of 40 are 40, 80, 120, 160 and 200 respectively.} \\ \\\text {Q4. Which of the following numbers have 15 as their factor? } \\ (i) \quad 15615 \quad (ii) \quad 123015 \\ \\\text {Sol. (i) 15 is a factor of 15615 because it is divisor of 15615} \\\text {(ii) 15 is a factor of 123015 because it is a divisor of 123015 } \\ \\\text {Q5. Which of the following numbers are divisible by 21 ? } \\ (i) \quad 21063 \quad (ii) \quad 20163 \\ \\\text {Sol. (i) } \quad 21063= 21000 + 63 \\ = 21 \times 1000 + 21 \times 3 \\ = 21 \times (1000 + 3 ) \\ = 21 \times 1003 \\ \text {Clearly, we can see 21 is factor of } 21 \times 1003 \\ \text {Hence, 21063 is divisible by 21} \\ \\\text {(ii) Sum of the digits of the given number 20163} \\ =2+0+1+6+3=12 \text { which is divisible by 3.} \\ \text {Hence, 20163 is divisible by 3.} \\\text {Again, a number is divisible by 7 if the difference between twice the one’s digit } \\ \text {and the number formed by the other digits is either 0 or multiple of 7} \\\Rightarrow \quad 2016 – (2 \times 3)=2016 – 6 = 2010 \text { which is not a multiple of 7.} \\ \text {Thus, 20163 is not divisible by 21} \\ \\ \end{array}