Math Olympiad Test: Real Numbers- 4 - Class 10 MCQ

Given number: 
Rationalising factor is the number which when multiplied with given number removes all surds.
∴ Rationalising factor = 

We have, 
Rationalising both terms, we get

Given number,

If the prime factorisation of denominator has power of 2, power of 5 or both, then the number should always be terminating decimals. So, it is a non-terminating and repeating decimal expansion.

π is an irrational number, which means that it is a real number that cannot be expressed by a simple fraction. That's because pi is what mathematicians call an "infinite decimal" — after the decimal point, the digits go on forever and ever.

HCF of (192, 240, 168) = 2 × 2 × 2 × 3 = 24
Number of rooms for participants in Mathematics, Physics and Biology respectively is= 192/24 = 8, 240/24 = 10 and 168/24 = 7
∴ Total minimum number of required rooms = 25

Let b = p₁​, p₂​, p₃​, p₄​.....pn​ where p_1​, p₂​, p_3​,.. are prime numbers which are necessarily not distinct,  
⇒ b² = (p₁​, p₂​, p₃​, p₄.....p_n​)(p₁​, p₂​, p₃​, p₄​.....p_n​) 
It is given that p divides b². 
From Fundamental theorem of Arithmetic, we know that every composite number can be expressed as product of unique prime numbers. 
This means p belongs to p₁​, p₂​, p₃​, ..p_n​ and is one of them.
Also, b = p₁​, p₂​, p₃​, p₄​...p_n​, thus p divides b.

987 = 3 × 7 × 47
So, the required number must be divisible by each one of 3, 7, 47.
553681→ (Sum of digits = 28, not divisible by 3)
555181→ (Sum of digits = 25, not divisible by 3)
555681 is divisible by 3, 7, 47.

L.C.M. (35, 32, 40) = 2⁵ × 5 × 7 = 1120
So, each should walk 1120 cm so that each can cover the same distance in complete steps.

First device beeps in every 30 min
Second device beeps in every 60 min
Third device beeps in every 90 min
Fourth device beeps in every 105 min
That means together they beep after how many minutes
We calculate LCM(30, 60, 90, 105)
30 = 2 × 3 × 5
60 = 2 × 2 × 3 × 5
90 = 2 × 3 × 3 × 5
105 = 3 × 5 × 7
So, LCM = 22 × 32 × 5 × 7 = 1260
⇒ ie, in 601260​ = 21hrs they beep together.
⇒ All beeped together after 12 noon at 21 hrs later. i.e, (12 + 21)hrs = 9 a.m.
∴  Next, they will beeped together at 9 a.m.

Let the number be x.
Then, according to question, 
⇒ 225x = 8 × 17 × 225 ⇒ x = 136