Question 24 of 27intermediate🔧 ApplyShort Answer3 marks

A number leaves a remainder of 3 when divided by 7, and another number leaves a remainder of 5 when divided by 7. What is the remainder when their sum, difference, and product are divided by 7?

Correct Answer

Given: Let the first number be 7a+37a + 3 and the second be 7b+57b + 5.

Sum:

(7a+3)+(7b+5)=7(a+b)+8=7(a+b+1)+1(7a + 3) + (7b + 5) = 7(a + b) + 8 = 7(a + b + 1) + 1

∴ Remainder when divided by 7 is 1.

Difference:

(7a+3)−(7b+5)=7(a−b)−2=7(a−b−1)+5(7a + 3) - (7b + 5) = 7(a - b) - 2 = 7(a - b - 1) + 5

∴ Remainder is 5.

Product:

(7a+3)(7b+5)=49ab+35a+21b+15=7(7ab+5a+3b+2)+1(7a + 3)(7b + 5) = 49ab + 35a + 21b + 15 = 7(7ab + 5a + 3b + 2) + 1

∴ Remainder is 1.

Exercise: Figure it Out (Section 3) | Q: 6 | (Page Page 20)
For More Understanding

Explanation

The chapter shows how algebraic expressions can represent numbers with specific remainders. A number leaving remainder 3 when divided by 7 can be written as 7a+37a + 3. The distributive property helps compute remainders of sums, differences, and products.

Solution Steps

  1. First number = 7a+37a + 3, Second number = 7b+57b + 5

  2. Sum = 7a+3+7b+5=7(a+b)+8=7(a+b+1)+17a + 3 + 7b + 5 = 7(a + b) + 8 = 7(a + b + 1) + 1

  3. Remainder of sum = 1

  4. Difference = 7a+3−7b−5=7(a−b)−2=7(a−b−1)+57a + 3 - 7b - 5 = 7(a - b) - 2 = 7(a - b - 1) + 5

  5. Remainder of difference = 5

  6. Product = (7a+3)(7b+5)=49ab+35a+21b+15(7a + 3)(7b + 5) = 49ab + 35a + 21b + 15

  7. =7(7ab+5a+3b+2)+1= 7(7ab + 5a + 3b + 2) + 1

  8. Remainder of product = 1

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