Real Numbers Test - 2

Euclid's division algorithm: Suppose the two positive numbers are 'a' and 'b' and are such that a > b. Then the H.C.F. of 'a' and 'b' can found by the following given below steps.
(a) Apply the division lemma to find 'q' and 'r' , where a = bq + r, b > 0. 0 ≤ r < b.
(b) If r = 0, then then H.C.F. is b. If r ≠ 0, then apply Euclid's lemma to find 'b' and 'r'.
(c) Continue steps (a) and (b) till r = 0. The divisor at this state will be H.C.F. (a, b). Also , H.C.F. ( a , b) = H.C.F (b , r).

Let the required number be x.
According to the question, when 38 is divided by x, we get remainder 2. This means (38 - 2) is completely divisible by x.
This means x is a factor of (38 - 2).
Similarly, x is a factor of (45 - 3) and (52 - 4).
Since we need to find the largest common factor of 36, 42 and 48, we will find the HCF of (36, 42, 48).
Now,
Step 1: Using Euclid`s division algorithm on 36 and 42, we get
42 = 36 <sup><sub>×</sub></sup> 1 + 6
36 = 6 <sup><sub>×</sub></sup> 6 + 0
HCF (36, 42) = 6
Step 2: Using Euclid`s division algorithm on 6 and 48, we get
48 = 6 ^_× 8 + 0
HCF (48, 6) = 6
Hence, HCF (36, 42, 48) = 6
∴The greatest number is 6.

The LCM of 2, 3, 4, 5 and 6 is 60. The number of books will be of the form 60K + 1. We put various values to K so as to make it divisible by 7.
Starting from K = 1,
60(1) + 1 = 61, which is not divisible by 7.
K = 2 ,
60(2) + 1 = 121, which is not divisible by 7.
K = 3
60(3) + 1 = 181, which is not divisible by 7.
K = 4
60(4) + 1 =241, which is not divisible by 7.
K = 5
60(5) + 1 = 301, which is divisible by 7.

The lengths of their march were 68 cm, 51 cm and 85 cm, respectively.
Therefore, the required distance travelled by them so as their steps were in the same line together = LCM of 68 cm, 51 cm, and 85 cm.
Now, the LCM of 68, 51, and 85 = 17 × 4 × 3 × 5 = 1020 cm = 10.2 m

According to question,
When we divide 3838 by the required number, it leaves a remainder 3.
Therefore, the number will completely divide 3835 (3838 - 3).
Also, when we divide 5123 by the required number, it leaves a remainder 13.
Therefore, the number will completely divide 5110 (5123 - 13 ).
The total number of t-shirts sold on Monday = HCF of 3835 and 5110
The HCF of 3835 and 5110 is 5.

As we know that the Aryan's age is divisible by 14.
Also, Aryan`s friend notices that Aryan's age is divisible by 3.
Therefore, the lowest possible age of Aryan = LCM of 14 and 3 = 42 years.

All the three tube lights glow at once at 10 a.m.
The time when they glow simultaneously again = LCM (60,48,72) seconds = 720 seconds = 12 mins.
Therefore, the time when three tube lights glow together again is 10:12:00 a.m.

(A) Natural numbers:
A set of counting numbers is called natural numbers.
N = {1,2,3,4,5,...}
Whole number :
A set of natural numbers along with zero is called whole numbers.
W= {0,1,2,3,4,5,....}
So, All natural numbers are whole numbers but all whole numbers are not natural numbers.
So, (A) is correct.
(B) Every integer, natural and whole number is a rational number (not irrational numbers) as they can be expressed in terms of p/q.
So, (B) is incorrect

(C) All rational and all irrational numbers make a collection of real numbers. It is denoted by the letter R .
So, (C) is correct.

(D) Let n – 1 and n be two consecutive positive integers.
Then their product is n (n – 1) = n² – n.
We know that every positive integer is of the form 2q or 2q + 1 for some integer q.
So, let n = 2q
So,
n² – n
= (2q)²– (2q)
= 4q² – 2q
= 2q (2q – 1)
= 2r [where r = q(2q – 1)]
n² - n is even and divisible by 2.
Let n = 2q + 1
So,
n² – n = (2q + 1) ²– (2q + 1)
= (2q + 1) ((2q + 1) – 1)
= (2q + 1) (2q)
= 2r[r = q (2q + 1)]
n² – n is even and divisible by 2 and not by 3.

(1) Every real number (W) can be represented by a numerical string of digits, possibly continuing forever. This is called the decimal representation (X) of the number. Integers, like 3, -2, 0, 42, -5,658 and 142,235 are already in their decimal representation.The decimal representation of many numbers, such as π, have no pattern to the sequence of digits. Rational numbers are those that can be written as the quotient of two whole numbers, either have a terminating decimal representation or have a repeating representation.

(2) The set of all integers (Y) and fractional numbers between them comprise the set of rational numbers (Z). The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2 (1.41421356..., the square root of 2, an irrational algebraic number). Included within the irrationals are the transcendental numbers, such as π (3.14159265...).