PRACTICE QUESTIONS ON NUMBERS FOR GRADE 10

Question 1 :

Prove that n² -n divisible by 2 for every positive integer n.

Solution :

n² -n   =  n(n - 1)

If n is odd, then n - 1 is even.

If n - 1 is odd, then n will be even.

The product of odd number and even number will be an even number.

Hence n² -n is divisible by 2.

Question 2 :

A milk man has 175 litres of cow’s milk and 105 litres of buffalow’s milk. He wishes to sell the milk by filling the two types of milk in cans of equal capacity. Calculate the following (i) Capacity of a can (ii) Number of cans of cow’s milk (iii) Number of cans of buffalow’s milk

Solution :

Since the capacity of two can will be equal, we have to find H.C.F.

175  =  5² x 7

105  =  5 x 3 x 7

H.C.F  =  5 x 7   =  35

Hence the capacity of each can is 35 liters.

(ii) (iii) Number of cans of Cow's milk

To find number of can of cows milk, we have to divide 175 by 35.

175/35  =  5

Hence 5 cans of cow's milk is available.

(iii) Number of cans of buffalow’s milk

To find number of can of cows milk, we have to divide 105 by 35.

105/35  =  3

Hence 3 cans of buffalow’s milk is available.

Question 3 :

When the positive integers a, b and c are divided by 13 the respective remainders are 9, 7 and 10. Find the remainder when a +2b + 3c is divided by 13.

Solution :

Division algorithm :

Divided  =  Divisor x quotient + Remainder

a  =  13q₁ + 9 -------(1)

b  =  13q₂ + 7

2b  =  26q₂ + 14  -------(2)

c  =  13q₃ + 10

3c  =  39q₃ + 30 -------(3)

(1) + (2) + (3)

a + 2b + 3c  =  13q₁ + 9 + 26q₂ + 14 + 39q₃ + 30

  =  13(q₁ + q₂ + q₃) + 53

The remainder is 53.

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