CH 1 – REAL NUMBERS
DAY – 1

Show that any positive odd integer is of form 41 + 1 or 4q + 3 where q is a positive integer.

Show that one and only one out of n, (n + 1) and (n + 2) is divisible by 3, where n is any positive integer.

Find the HCF of numbers 134791, 6341, and 6339 by Euclid’s division algorithm.

Find HCF and LCM of 13 and 17 by prime factorization method.

Using Euclid’s division algorithm, find whether the pair of numbers 847, and 2160 are coprime or not.

Find the HCF of 255 and 867 by Euclid’s division algorithm.

By using Euclid’s algorithm, find the largest number which divides 650 and 1170.

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

Two tankers contain 850 liters and 680 liters of petrol. Find the maximum capacity of a container that
can measure the petrol of each tankerthe exact number of times.

Find the LCM of 96 and 360 by using the fundamental theorem of arithmetic.

Find LCM of numbers whose prime factorization are expressible as 3 × 5^{2} and 3^{2} × 7^{2}.

HCF and LCM of two numbers is 9 and 459 respectively. If one of the numbers is 27, find the other number.

Find the HCF (865, 255) using Euclid’s division lemma.

The decimal expansion of the rational number 43/2^4*5^3 will terminate after how many places of decimals?

Find the largest number that will divide 398, 436 and 542 leaving remainders 7, 11, and 15 respectively.
DAY – 2

Express 98 as a product of its primes.

Express 98 as a product of its primes.

Find the largest number which divides 70 and 125 leaving remainder 5 and 8 respectively.

Prove that 2 + 3√5 is an irrational number.

Check whether 4n can end with the digit 0 for any natural number n.

Can two numbers have 15 as their HCF and 175 as their LCM?

Explain why (17 × 5 × 11 × 3 × 2 + 2 × 11) is a composite number.

Show that 3√7 is an irrational number.

Find the prime factorization of the denominator of a rational number expressed as 6.12¯ in simplest form.

Prove that √5 is irrational and hence show that 3 + √5 is also irrational.

Prove that 3 + 2√3 is an irrational number.

Three bells toll at intervals of 9, 12, 15 minutes respectively. If they start tolling together, after what time will they next toll together?

Three alarm clocks ring at intervals of 4, 12 and 20 minutes respectively. If they start ringing together, after how much time will they next ring together?

In a school, there are two Sections A and B of class X. There are 48 students in Section A and 60 students in Section B. Determine the least number of books required for the library of the school so that the books can be distributed equally among all students of each Section.

Dudhnath has two vessels containing 720 ml and 405 ml of milk respectively. Milk from these containers is poured into glasses of equal capacity to their brim. Find the minimum number of glasses that can be filled.

Find the HCF and LCM of 306 and 657 and verify that LCM × HCF = Product of the two numbers.

Amita, Sneha, and Raghav start preparing cards for all persons of an old age home. In order to complete one card, they take 10, 16, and 20 minutes respectively. If all of them started together, after what time will they start preparing a new card together?

The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form, p/q what can you say about the prime factors of q?
(i) 43.123456789
(ii) 0.120120012000120000. . .

Check whether 6^{n} can end with the digit 0 for any natural number n.

What is the HCF of the smallest prime number and the smallest composite number?
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