Exploring the Real Number System

Introduction to "Method of finding HCF of Two Numbers by using Euclid’s Division Algorithm."

Using Euclid's division algorithm, find the H.C.F. of 13 and 25.

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

Use Euclid’s algorithm to find the HCF of 4052 and 12576.

Find the largest number which divides 245 and 1029 leaving remainder 5 in each case.

If the HCF of 65 and 117 is expressible in the form 65m – 117, then find the value of ‘m’.

A sweet seller has 420 kajubarfis and 130 badambarfis. She wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray. What is the number of that can be placed in each stack for this purpose?

Introduction to "Finding Properties of numbers by Euclid’s Division Lemma."

Show that every positive even integer is of the form 2q, and every positive add Integer is of the form 2q + 1, where q is some integer.

Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer.

Show that(n^2 – 1) is divisible by 8, if n is an odd positive integer.

Prove that, if both x and y are positive odd integers, then (x^2 + y^2) is an even integer but not divisible by 4.

Finding properties of numbers - Continuation

show that the square of any positive odd integer is of the form 4m + 1 for some integer m.

Show that the cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3, for some integer m.

Introduction to "Fundamental theorem of Arithmetic:"

Explain why ( 3 × 5 × 7) + 7 is a composite number ?

Consider the numbers 4^n, where n is a natural number. Check whether there is any value of n for which 4^n ends with the digit zero.

Find the LCM and HCF of 6 and 20 by the prime factorisation method.

Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM.

Find the LCM and HCF of 336 and 54 and verify that LCM × HCF = product of the two numbers.

The Fundamental Theorem of Arithmetic - Applications

If two positive integers p and q are written as p = a^2b^3 and q = a^3b, where a and b are prime numbers then verify. LCM (p, q) HCF (p, q) = p x q.

Write the HCF and LCM of smallest odd composite number and the smallest odd prime number. If an odd number p divides q^2, then will it divide q^3 also? Explain.

Amita, Suneha and Raghav start preparing cards for greeting each person of an old age home on new year. 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?

Why do you think there is a need to show elders that the young generation cares for them and remembers the contribution made by them in the prime of their life?

Find the greatest number of six digits exactly divisible by 18, 24 and 36.

Introduction to "Revisiting Irrational Numbers:"

If p is a prime number, then prove that square root p is an irrational.

Prove that square root 2  is irrational.

Prove that square root 3  is irrational.

Revisiting Irrational Numbers - Continuation

Show that 5 - (square root 3) is irrational.

Show that 3(square root 2) is an irrational.

Show that there is no positive integer n, for which "square root(n - 1) + square root(n + 1)" is rational.

Introduction to “Revisiting Rational Numbers and Their Decimal Expansions :”

Write whether the rational number 7/75 will have a terminating decimal expansion or a non-terminating repeating decimal.

Write whether the given expression(combination of rationals and irrationals), on simplification gives a rational or an Irrational number.

Express the number 0.3(178) bar  in the form of rational number a/b.

Use Euclid’s division algorithm to

find the HCF of:

(i) 135 and 225

(ii) 196 and 38220

(iii)867and255

Use Euclid’s division algorithm to

find the HCF of:

(ii) 196 and 38220

Use Euclid’s division algorithm to

find the HCF of:

(iii)867and255

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

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?

Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

[Hint: Let x be any positive integer then it is of the form 3q, 3q + 1 or 3q + 2. Now square each of these and show that they can be rewritten in the form 3m or 3m + 1]

Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.