Real Numbers MATHS - Real Numbers

For some integer q, every odd integer is of the form:

The product of two irrational numbers is an irrational number (True/ False).

Two numbers have 12 as their HCF and 350 as their LCM (True/ False).

Every even integer is of the form 2m, where m is an integer (True/ False).

Every odd integer is of the form 2m - 1, where m is an integer (True/ False).

The product of any three consecutive natural number is divisible by 6 (True/ False).

If a and b are relatively prime numbers, then what is their LCM?

What is a lemma?

Given that HCF (306. 657) = 9, find LCM (306, 657).

Determine the prime factorisation of the following positive integer:
20570

Find the LCM and HCF of the following integer by applying the prime factorisation method.
17, 23 and 29

Define HCF of two positive integers and find the HCF of the following pairs of numbers:
475 and 495

What is a composite number?

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?

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

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

Show that the square of any positive integer cannot be of the form 6m + 2 or 6m + 5 for any integer m.

Determine the number nearest to 110000 but greater than 100000 which is exactly divisible by each of 8, 15 and 21.

In a morning walk three persons step off together, their steps measure 80cm, 85cm and 90cm respectively. What is the minimum distance each should walk so that he can cover the distance in complete steps?

Using Euclid’s division algorithm, find the largest number that divides 1251, 9377 and 15628 leaving remainders 1, 2 and 3 respectively.

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.