Class 6 Playing with Numbers Exercise 2.5-2

\begin{array}{l} \text {Q7. Test the divisibility of the following numbers by 11 :} \\ (i) \quad 5335 \\ (ii) \quad 70169803 \\ (iii) \quad 10000001 \\ \\\text {Sol. We know that a natural number is divisible by 11, } \\ \text {if the difference of the sim of its digits in odd places and sum } \\ \text {of its digits in even places is either 0 or a multiple of 11.} \\\text {(i) Given number } =5335 \\ \text {Sum of digits at odd places } = 5+3=8 \\ \text {Sum of digits at even places } = 3+5=8 \\ \text {Difference of two sums } = 8-8=0 \\ \therefore \quad 5335 \text { is divisible by 11.} \\ \\\text {(ii) Given number } =70169803 \\ \text {Sum of digits at odd places } = 3+8+6+0=17 \\ \text {Sum of digits at even places } = 0+9+1+7=17 \\ \text {Difference of two sums } = 17 -17=0 \\ \therefore \quad 70169803 \text { is divisible by 11.} \\ \\\text {(iii) Given number } =10000001 \\ \text {Sum of digits at odd places } = ! +0+0+0=1 \\ \text {Sum of digits at even places } = 0+0+0+0=1 \\ \text {Difference of two sums } = 1 -1=0 \\ \therefore \quad 10000001 \text { is divisible by 11.} \\ \\\text {Q8. In each of the following numbers, replace * by the smallest number to make it divisible by 3 :} \\ (i) \quad 75 * 5 \\ (ii) \quad 35 * 64 \\ (iii) \quad 18 * 71 \\ \\\text {Sol. (i) 75 * 5 } \\ \text {Sum of digits } = 7+5+5 = 17 \\ \text {We know that 18 is multiple of 3 which is greater than 17.} \\ \therefore \text {The smallest required number } = 18 – 17 = 1 \\ \text {Hence, the smallest required number is 1.} \\ \\(ii) \quad 35 * 64 \\ \text {Sum of digits } = 3+5+6+4 = 18 \\ \text {We know that 18 divisible by 3.} \\ \text {Hence, the smallest required number is 0.} \\ \\(iii) \quad 18 * 71 \\ \text {Sum of digits } = 1+8+7+1= 17 \\ \text {We know that 18 is multiple of 3 which is greater than 17.} \\ \therefore \text {The smallest required number } = 18 – 17 = 1 \\ \text {Hence, the smallest required number is 1.} \\ \\\text {Q9. In each of the following numbers, replace * by the smallest number to make it divisible by 9 :} \\ (i) \quad 67 * 19 \\ (ii) \quad 66784 * \\ (iii) \quad 538 * 8 \\ \\\text {Sol. (i) 67 * 19 } \\ \text {Sum of digits } = 6+7+1+9 = 23 \\ \text {We know that 27 is multiple of 9 which is greater than 23.} \\ \therefore \text {The smallest required number } = 27 -23 = 4 \\ \text {Hence, the smallest required number is 4.} \\ \\(ii) \quad 66784 * \\ \text {Sum of digits } = 6+6+7+8+4= 31 \\ \text {We know that 36 is multiple of 9 which is greater than 31.} \\ \therefore \text {The smallest required number } = 36 – 31 = 5 \\ \text {Hence, the smallest required number is 5.} \\ \\(iii) \quad 538 * 8 \\ \text {Sum of digits } = 5+3+8+8= 24 \\ \text {We know that 27 is multiple of 9 which is greater than 24.} \\ \therefore \text {The smallest required number } = 27 – 24 = 3 \\ \text {Hence, the smallest required number is 3.} \\ \\\text {Q10 . In each of the following numbers, replace * by the } \\ \text {smallest number to make it divisible by 11 :} \\ (i) \quad 86 * 72 \\ (ii) \quad 467 * 91 \\ (iii) \quad 9 * 8071 \\ \\\text {Sol. We know that a natural number is divisible by 11, } \\ \text {if the difference of the sim of its digits in odd places and sum } \\ \text {of its digits in even places is either 0 or a multiple of 11.} \\\text {Lets assume that the required number be x} \\(i) \quad 86 * 72 \\ \text {Sum of digits at odd places } = 2+x+8=x +10 \\ \text {Sum of digits at even places } = 7+6=13 \\ \text {Difference of two sums } = x+10-13=x-3 \\ \text {As per rule, } x-3 = 0 \quad [\because \text {Missing number is a single digit.}] \\ \Rightarrow \quad x = 3 \text {Hence, the smallest required number is 3.} \\ \\(ii) \quad 467 * 91 \\ \text {Sum of digits at odd places } = 1+x+6=x + 7 \\ \text {Sum of digits at even places } = 9+7+4=20 \\ \text {Difference of two sums } = 20 – x -7=13 – x \\ \text {As per rule, } 13-x = 11 \quad [\because \text {Missing number is a single digit.}] \\ \Rightarrow \quad x = 2 \text {Hence, the smallest required number is 2.} \\ \\(iii) \quad 9 * 8071 \\ \text {Sum of digits at odd places } = 1+0+x=x + 1 \\ \text {Sum of digits at even places } = 7+8+9=24 \\ \text {Difference of two sums } = 24 – x -1=23 – x \\ \text {As per rule, } 23-x = 22 \quad [\because \text {22 is a multiple of 11.}] \\ \Rightarrow \quad x = 1 \text {Hence, the smallest required number is 1.} \\ \\\text {Q11. Given an example of a number which is divisible by } \\ \text {(i) 2 but not by 4 } \\ \text {(ii) 3 but not by 6 } \\ \text {(iii) 4 but not by 8 } \\ \text {(iv) both 4 and 8 but not by 32 } \\ \\\text {(i) A number which is divisible by 2 but not by 4 is 10.} \\ \\\text {(ii) A number which is divisible by 3 but not by 6 is 15.} \\ \\ \text {(iii) A number which is divisible by 4 but not by 8 is 28.} \\ \\\text {(iv) A number which is divisible by both 4 and 8 but not by 32 is 48.} \\ \\\text {Q12. Which of the following statements are true? } \\ \text {(i) If a number is divisible by 3, it must be divisible by 9 } \\ \text {(ii) If a number is divisible by 9, it must be divisible by 3 } \\ \text {(iii) If a number is divisible by 4, it must be divisible by 8.} \\ \text {(iv) If a number is divisible by 8, it must be divisible by 4.} \\ \text {(v) A number is divisible by 18, if it is divisible by both 3 and 6.} \\ \text {(vi) If a number is divisible by both 9 and 10, it must be divisible by 90.} \\ \text {(vii) If a number exactly divides the sum of two numbers, } \\ \text {it must exactly divide the numbers separately.} \\ \text {(viii) If a number divides three numbers exactly, it must divide their sum exactly.} \\ \text {(ix) If two numbers are co-prime, at least one of them must be a prime number.} \\ \text {(x) The sum of two consecutive odd numbers is always divisible by 4 } \\ \\\text {Sol. (i) False. The number 15 is divisible by 3 but not by 9.} \\ \\\text {(ii) True. The number 18 is divisible by 9 and by 3 } \\ \\\text {(iii) False. 12 is divisible by 4 but not by 8.} \\ \\\text {(iv) True. The number 16 is divisible by 8 and by 4} \\ \\\text {(v) False. 24 is divisible by both 3 and 6 but not divisible by 18.} \\ \\\text {(vi) True. 90 is divisible by both 9 and 10.} \\ \\\text {(vii) False. The number 10 divides the sum of 18 and 2 but 10 does not divide 18 or 2} \\ \\\text {(viii) True. 2 divides the number 4,6 and 8 and the sum 18 exactly.} \\ \\ \text {(ix) False. The numbers 4 and 9 are co-primes which are composite numbers. } \\ \\\text {(x) True. The sum 3+5=8 is divisible by 4.} \\\end{array}