Math Olympiad Test: Real Numbers- 3 - Class 10 MCQ

We have, n = 9q + 7, where q is the quotient.
⇒ 3n = 27q + 21 (Multiply by 3)
⇒ 3n – 1 = 27q + 20 = 9(3q + 2) + 2
So, remainder is 2, when (3n – 1) is divided by 9.

If the prime factorisation of q in rational number p/q is of the form of 2ⁿ × 5^m, where m and n are non-negative integers, then the number has a terminating decimal expansion.

Given number is 
Let x = 
Multiplying (i) by 10, we get 10x = 
Subtracting (i) from (ii), we get

If 
we need to find the value of a and b
Rationalize the left-hand side of given expression,

Compare both the sides,
so, we get the values a = 7 and b = 4

LCM (306, 954, 1314)

Let n be a natural number, then (n + 1) will be a consecutive natural number.
⇒ Their product = n(n + 1)
So, the product would always be an even number.
If n is odd, then n + 1 would be even, their product is even.
If n is even, then n + 1 would be odd, again their product is even.

L.C.M. of 3, 6, 9 is 18.

Descending order is, 

100^th root of 

We have,
 
On rationalising all the terms, we get
√2 − 1+ √3 − √2 + √4 − √3 + √5 − √4 + √6 − √5 + √7 − √6 + √8 − √7 + √9 − √8
= √9 − 1 = 3 – 1 = 2