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# R.D. Sharma solutions 10 Mathematics chapter 1 Real Numbers

## Chapter 1 - Real Numbers

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Prove that the product of two consecutive positive integers is divisible by 2.

If a and b are two odd positive integers such that a > b, then prove that one of the two numbers (a+b)/2and(a-b)/2 is odd and the other is even.

Show that the square of an odd positive integer is of the form 8q + 1, for some integer q.

Show that any positive odd integer is of the form 6q + 1 or, 6q + 3 or, 6q + 5, where q is some integer.

Prove that the square of any positive integer is of the form 3m or, 3m + 1 but not of the form 3m +2.

Prove that the square of any positive integer is of the form 4q or 4q + 1 for some integer q.

Prove that the square of any positive integer is of the form 5q, 5q + 1, 5q + 4 for some integer q.

Prove that if a positive integer is of the form 6q + 5, then it is of the form 3q + 2 for some integer q, but not conversely.

Prove that the square of any positive integer of the form 5q + 1 is of the same form.

Prove that the product of three consecutive positive integer is divisible by 6.

For any positive integer n , prove that n3 − n divisible by 6.

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Define HOE of two positive integers and find the HCF of the following pair of numbers:

32 and 54

Define HOE of two positive integers and find the HCF of the following pair of numbers:

18 and 24

Define HOE of two positive integers and find the HCF of the following pair of numbers:

70 and 30

Define HOE of two positive integers and find the HCF of the following pair of numbers:

56 and 88

Define HOE of two positive integers and find the HCF of the following pair of numbers:

475 and 495

Define HOE of two positive integers and find the HCF of the following pair of numbers:

75 and 243

Define HOE of two positive integers and find the HCF of the following pair of numbers:

240 and 6552

Define HOE of two positive integers and find the HCF of the following pair of numbers:

155 and 1385

Define HOE of two positive integers and find the HCF of the following pair of numbers:

100 and 190

Define HOE of two positive integers and find the HCF of the following pair of numbers:

105 and 120

Using Euclid's division algorithm, find the H.C.F. of 135 and 225

Using Euclid's division algorithm, find the H.C.F. of 196 and 38220

Find the HCF of the following pairs of integers and express it as a linear combination of 963 and 657.

Find the HCF of the following pairs of integers and express it as a linear combination of 592 and 252.

Find the HCF of the following pairs of integers and express it as a linear combination of 506 and 1155.

Find the HCF of the following pairs of integers and express it as a linear combination of 1288 and 575.

Express the HCF of 468 and 222 as 468x + 222y where x, y are integers in two different ways.

If the HCF of 408 and 1032 is expressible in the form 1032 m − 408 × 5, find m.

If the HCF of 657 and 963 is expressible in the form 657x + 963y − 15, find x.

An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

Find the largest number which divides 615 and 963 leaving remainder 6 in each case.

Find the greatest number which divides 285 and 1249 leaving remainders 9 and 7 respectively.

Find the largest number which exactly divides 280 and 1245 leaving remainders 4 and 3, respectively.

What is the largest number that divides 626, 3127 and 15628 and leaves remainders of 1, 2 and 3 respectively.

Find the greatest number that will divide 445, 572 and 699 leaving remainders 4, 5 and 6 respectively.

Find the greatest number which divides 2011 and 2623 leaving remainders 9 and 5 respectively.

The length, breadth and height of a room are 8 m 25 cm, 6 m 75 cm and 4 m 50 cm, respectively. Determine the longest rod which can measure the three dimensions of the room exactly.

105 goats, 140 donkeys and 175 cows have to be taken across a river. There is only one boat which will have to make many trips in order to do so. The lazy boatman has his own conditions for transporting them. He insists that he will take the same number of animals in every trip and they have to be of the same kind. He will naturally like to take the largest possible number each time. Can you tell how many animals went in each trip?

15 pastries and 12 biscuit packets have been donated for a school fete. These are to be packed in several smaller identical boxes with the same number of pastries and biscuit packets in each. How many biscuit packets and how many pastries will each box contain?

A mason has to fit a bathroom with square marble tiles of the largest possible size. The size of the bathroom is 10 ft. by 8 ft. What would be the size in inches of the tile required that has to be cut and how many such tiles are required?

Two brands of chocolates are available in packs of 24 and 15 respectively. If I need to buy an equal number of chocolates of both kinds, what is the least number of boxes of each kind I would need to buy?

144 cartons of Coke Cans and 90 cartons of Pepsi Cans are to be stacked in a Canteen. If each stack is of the same height and is to contain cartons of the same drink, what would be the greatest number of cartons each stack would have?

During a sale, colour pencils were being sold in packs of 24 each and crayons in packs of 32 each. If you want full packs of both and the same number of pencils and crayons, how many of each would you need to buy?

A merchant has 120 liters of oil of one kind, 180 liters of another kind and 240 liters of third kind. He wants to sell the oil by filling the three kinds of oil in tins of equal capacity. What should be the greatest capacity of such a tin?

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