Question 6 - Figure it out - Page 132, 133, 134 - Chapter 5 Class 8 - Number Play (Ganita Prakash)

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Question 6 If 3p7q8 is divisible by 44, list all possible pairs of values for p and q.If a number is divisible by 44, it means It is divisible by 4 (must be even) It is divisible by 11 (sum of digits is divisible by 9). Note: We choose 4 & 11 because they have no common factors Divisibility by 4 Number is 3p7q8 If the number is divisible by 4, last two digits q8 must me a multiple of 4 So, possible last two digits can be 08, 28, 48, 68, 88 Thus, q can be 0, 2, 4, 6, 8 Divisibility by 11 3p7q8 Sum of digits at Odd Places = 3 + 7 + 8 = 18 Sum of digits at Even Places = p + q Difference = 18 − (p + q) Thus, 18 − (p + q) is divisible by 11 Which is only possible if 18 − (p + q) = 0 or 18 − (p + q) = 11 Thus, p + q = 18 or p + q = 7 But, for p + q = 18 Both p & q should be 9 This is not possible as q can be 0, 2, 4, 6, 8 Thus, p + q = 7 And, q can be 0, 2, 4, 6, 8 So, our possible pairs are If 𝒒=𝟎,𝒑=𝟕. If 𝒒=𝟐,𝒑=𝟓. If 𝒒=𝟒,𝒑=𝟑. If 𝒒=𝟔,𝒑=𝟏. Thus, our answer is Pairs (𝑝,𝑞) are (𝟕,𝟎),(𝟓,𝟐),(𝟑,𝟒),(𝟏,𝟔).