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Ex 3.3, 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 11 (say, yes or no):Divisibility by 2 If last digit is 0, 2, 4, 6, 8 Divisibility by 3 If Sum of digit is divisible by 3 Divisible by 4 If last 2 digits are divisible by 4 Divisibility by 5 If its last digit is either 0 or 5 Divisibility by 6 If its divisible by 2 & 3 Divisible by 8 If last 3 digits are Divisible by 8 Divisibility by 9 If sum of its digits is divisible by 9 Divisibility by 10 If last digit is 0 Divisible by 11 If Sum of digits at odd place − Sum of digits at even Place = 0 or Number divisible by 11 Divisibility by 2 Since last digit is 0, It is divisible by 2 Divisibility by 3 Sum of digits = 9 + 9 + 0 = 18 Since 18 is divisible by 3 990 is divisible 3 Divisiblility by 4 Last 2 digits = 90 Since 90 is not divisible by 4 990 is not divisible by 4 Divisibility by 5 Since last digit is 0 It is divisible by 5 Divisibility by 6 Since it is divisible by 2 & 3 It is divisible by 6 Divisiblity by 8 Last 3 digits = 990 ∴ 990 is not divisible by 8 Divisibility by 9 Sum of digits = 9 + 9 + 0 = 18 Since 18 is divisible by 9 990 is divisible by 9 Divisibility by 10 Since last digit is 0 It is divisible by 10 Divisible by 11 990 Sum of odd digits = 0 + 9 = 9 Sum of even digits = 9 Difference = 9 − 9 = 0 Since difference is 0 It is divisible by 11 Divisibility by 2 Last digit is 6, so divisible by 2 Divisibility by 3 Sum of digits = 1 + 5 + 8 + 6 = 20 Since 20 is not divisible by 3 ∴ 1586 is not divisible by 3 Divisiblility by 4 Last 2 digits = 86 Since 86 is not divisible by 4 ∴ 1586 is not divisible by 4 Divisibility by 5 Last digit is 6 So 1586 is not divisible by 5 Divisibility by 6 1586 is not divisible by 3 ∴ 1586 is not divisible by 6 Divisiblity by 8 Last 3 digits = 586 Since 586 is not divisible by 8 ∴1586 is not divisible by 8 Divisibility by 9 Sum of digits = 1 + 5 + 8 + 6 = 20 Since 20 is not divisible by 9 1586 is not divisible by 9 Divisibility by 10 Last digit is 6 So 1586 is not divisible by 10 Divisible by 11 1586 Sum of odd digits = 6 + 5 = 11 Sum of even digits = 1 + 8 = 9 Difference = 11 − 9 = 2 Since 2 is not divisible by 11 So 1586 is not divisible by 11 Divisibility by 2 Last digit = 5 ∴ It is not divisible by 2 Divisibility by 3 Sum of digits = 2 + 7 + 5 = 14 Since 14 is not divisible by 3 275 is not divisible by 3 Divisiblility by 4 Last 2 digits = 75 Since 75 is not divisible by 4 275 is not divisible by 4 Divisibility by 5 Last digit is 5 So divisible by 5 Divisibility by 6 Since 275 is not divisible by 2 & 3 It is not divisible by 6 Divisiblity by 8 Last 3 digits = 275 ∴ 275 is not divisible by 8 Divisibility by 9 Sum of digits = 2 + 7 + 5 = 14 Since 14 is not divisible by 9 ∴ 275 is not divisible by 9 Divisibility by 10 Last digit = 5 So not divisible by 10 Divisible by 11 275 Sum of odd digits = 5 + 2 = 7 Sum of even digits = 7 Difference = 0 Since difference is 0 ∴ 275 is divisible by 11 Divisibility by 2 Last digit = 6 So divisible by 2 Divisibility by 3 Sum of digits = 6 + 6 + 8 + 6 = 26 Since 26 is not divisible by 3 275 is not divisible by 3 Divisiblility by 4 Last two digits = 86 Since 86 is not divisible by 4 6686 is not divisible by 4 Divisibility by 5 Last digit = 6 So not divisible by 5 Divisibility by 6 Since 6686 is not divisible by 3 6686 it is not divisible by 6 Divisiblity by 8 Last three digits = 686 Since 686 is not divisible by 8 ∴ 6686 is not divisible by 8 Divisibility by 9 Sum of digits = 26 Since 26 is not divisible by 9 ∴ 6686 is not divisible by 10 Divisibility by 10 Last digit = 6 So not divisible by 10 Divisibility by 11 6686 Sum of odd digits = 6 + 6 = 12 Sum of even digits = 8 + 6 = 14 Difference = 14 − 12 = 2 Since 2 is not divisible by 11 ∴ 6686 is not divisible by 11 Divisibility by 2 Last digit = 0 So divisible by 2 Divisibility by 3 Sum of digits = 6 + 3 + 9 + 2 + 1 + 0 = 21 Since 21 is divisible by 3 639210 is divisible by 3 Divisiblility by 4 Last two digits = 10 Since 10 is not divisible by 4 ∴ 639210 is not divisible by 4 Divisibility by 5 Last digit = 0 So, divisible by 5 Divisibility by 6 Since 639210 is divisible by both 2 & 3 It is also divisible by 6 Divisiblity by 8 Last 3 digits = 210 Since 210 is not divisible by 8 ∴ 639210 is not divisible by 8 Divisibility by 9 Sum of digits = 6 + 3 + 9 + 2 + 1 + 0 = 21 Since 21 is not divisible by 9 ∴ 639210 is not divisible by 9 Divisibility by 10 Last digit = 0 So divisible by 10 Divisibility by 11 639210 Sum of odd digits = 0 + 2 + 3 = 5 Sum of even digits = 1 + 9 + 6 = 16 Difference = 16 − 5 = 11 Since difference is 11 ∴ 639210 is divisible by 11 Divisibility by 2 Last digit = 4 ∴ It is divisible by 2 Divisibility by 3 Sum of digits = 4 + 2 + 9 + 7 + 1 + 4 = 27 Since 27 is divisible by 3 429714 is divisible by 3 Divisibilility by 4 Last two digits = 14 Since 14 is not divisible by 4 ∴ 429714 is not divisible by 4 Divisibility by 5 Last digit = 4 So 429714 is not divisible by 5 Divisibility by 6 Since it is divisible by both 2 & 3 It is also divisible by 6 Divisibility by 8 Last 3 digits = 714 Since 714 is not divisible by 8 ∴ 429714 is not divisible by 8 Divisibility by 9 Sum of digits = 4 + 2 + 9 + 7 + 1 + 4 = 27 Since 27 is divisible by 9 ∴ 429714 is divisible by 9 Divisibility by 10 Last digit = 4 So 429714 is not divisible by 10 Divisibility by 11 429714 Sum of odd digits = 2 + 7 + 4 = 13 Sum of even digits = 4 + 9 + 1 = 14 Difference = 14 − 13 = 1 Since 1 is not divisible by 11 ∴ 429714 is not divisible by 11 Divisibility by 2 Last digit is 6, so divisible by 2 Divisibility by 3 Sum of digits = 2 + 8 + 5 + 6 = 21 Since 21 is divisible by 3 ∴ 2856 is divisible by 3 Divisiblility by 4 Last 2 digits = 56 Since 56 is divisible by 4 ∴ 2856 is divisible by 4 Divisibility by 5 Last digit is 6 So 1586 is not divisible by 5 Divisibility by 6 2856 is divisible by 2 & 3 ∴ 2856 is divisible by 6 Divisiblity by 8 Last 3 digits = 856 Since 856 is divisible by 8 ∴2856 is divisible by 8 Divisibility by 9 Sum of digits = 2 + 8 + 5 + 6 = 21 Since 21 is not divisible by 9 2856 is not divisible by 9 Divisibility by 10 Last digit is 6 So 2856 is not divisible by 10 Divisible by 11 2856 Sum of odd digits = 8 + 6 = 14 Sum of even digits = 2 + 5 = 7 Difference = 14 − 7 = 7 Since 7 is not divisible by 11 So 2856 is not divisible by 11 Divisibility by 2 Last digit = 0 So divisible by 2 Divisibility by 3 Sum of digits = 3 + 0 + 6 + 0 = 9 Since 9 is divisible by 3 3060 is divisible by 3 Divisiblility by 4 Last two digits = 60 Since 60 is divisible by 4 ∴ 3060 is divisible by 4 Divisibility by 5 Last digit = 0 So, divisible by 5 Divisibility by 6 Since 3060 is divisible by both 2 & 3 It is also divisible by 6 Divisiblity by 8 Last 3 digits = 060 = 60 Since 3060is not divisible by 8 ∴ 639210 is not divisible by 8 Divisibility by 9 Sum of digits = 3 + 0 + 6 + 0 = 9 Since 9 is divisible by 9 ∴ 3060 is divisible by 9 Divisibility by 10 Last digit = 0 So divisible by 10 Divisibility by 11 3060 Sum of odd digits = 0 + 0 = 0 Sum of even digits = 3 + 6 = 9 Difference = 9 – 0 = 9 Since difference is not 0 nor multiple of 11 ∴ 3060 is not divisible by 11 Divisibility by 2 Last digit = 9 ∴ It is not divisible by 2 Divisibility by 3 Sum of digits = 4 + 0 + 6 + 8 + 3 + 9 = 30 Since 30 is divisible by 3 ∴ It is divisible by 3 Divisiblility by 4 Last 2 digits = 39 Since 39 is not divisible by 4 406839 is not divisible by 4 Divisibility by 5 Last digit is 9 ∴ It is not divisible by 5 Divisibility by 6 Since 406839 is not divisible by 3 It is not divisible by 6 Divisiblity by 8 Last 3 digits = 839 ∴ 406839 is not divisible by 8 Divisibility by 9 Sum of digits = 4 + 0 + 6 + 8 + 3 + 9 = 30 Since 30 is not divisible by 9 ∴ It is not divisible by 9 Divisibility by 10 Last digit = 10 ∴ It is not divisible by 10 Divisible by 11 406839 Sum of odd digits = 9 + 8 + 0 = 17 Sum of even digits = 3 + 6 + 4 = 13 Difference = 4 Since difference is not 0 nor multiple of 11 ∴ It is divisible by 11

Ex 3.3

Ex 3.3, 1
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Ex 3.3, 2 (a) Important

Ex 3.3, 2 (b)

Ex 3.3, 2 (c) Important

Ex 3.3, 2 (d)

Ex 3.3, 2 (e)

Ex 3.3, 2 (f) Important

Ex 3.3, 2 (g)

Ex 3.3, 2 (h) Important

Ex 3.3, 2 (i)

Ex 3.3, 2 (j) Important

Ex 3.3, 3 (a) Important

Ex 3.3, 3 (b)

Ex 3.3, 3 (c) Important

Ex 3.3, 3 (d)

Ex 3.3, 3 (e) Important

Ex 3.3, 3 (f)

Ex 3.3, 3 (g) Important

Ex 3.3, 3 (h)

Ex 3.3, 3 (i) Important

Ex 3.3, 3 (j)

Ex 3.3, 4 (a)

Ex 3.3, 4 (b) Important

Ex 3.3, 4 (c)

Ex 3.3, 4 (d) Important

Ex 3.3, 4 (e) Important

Ex 3.3, 4 (f)

Ex 3.3, 5 Important

Ex 3.3, 6 Important

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.