# Mathematics Questions and Answers – Real Numbers – Arithmetic Fundamental Theorem

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This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Real Numbers – Arithmetic Fundamental Theorem”.

1. Which of the following is a composite number?
a) 2
b) 3
c) 9
d) 7

Explanation: A prime number has two factors the number itself and 1. In case of 9 there are three factors i.e. 3 × 3 × 1. Hence, it is not a prime number.

2. Which of the following is a prime number?
a) 31
b) 52
c) 21
d) 32

Explanation: A prime number has two factors the number itself and 1. In case of 31 there are two factors i.e. 31 and 1. Hence, it is a prime number.

3. The fundamental theorem of arithmetic states that, every composite number can be factorized as product of primes and this factorization is unique.
a) False
b) True

Explanation: Let us consider a composite number, say 25
25 can be factorized as 5 × 5 × 1. This factorization is unique for 25 and no other number can be represented in the same manner.
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4. The least common multiple of 135 and 24 is _________
a) 90
b) 360
c) 240
d) 1080

Explanation: 135 can be written as 3 × 3 × 3 × 5 × 1 and 24 can be written as 2 × 2 × 2 × 3 × 1.
LCM is the product of greatest power of each prime factor involved in the numbers.
Therefore, LCM = 2 × 2 × 2 × 3 × 3 × 3 × 5 = 1080

5. The highest common factor of 21 and 90 is _________
a) 3
b) 2
c) 1
d) 4

Explanation: 21 can be written as 3 × 7 × 1 and 90 can be written as 3 × 3 × 2 × 5.
HCF is the product of smallest power of each prime factor involved in the numbers.
Therefore, HCF = 3 × 1 = 3

6. The LCM of two numbers is 7991 and the two numbers are 61 and 131. What will be their HCF?
a) 2
b) 1
c) 3
d) 4

Explanation: For two numbers a and b, we know that
(a × b) = HCF of (a, b) × LCM of (a, b)
Here a = 61 and b = 131, and LCM is 7991
61 × 131 = HCF × 7991
HCF = $$\frac {7991}{7991}$$ = 1

7. What will be the largest number that divides 100 and 25, and leaves 3 as remainder in each case?
a) 7
b) 5
c) 1
d) 4

Explanation: The required number divides (100-3) i.e. 97 and (25-3) i.e. 22 exactly.
Now, 97 = 97 × 1 and 22 = 2 × 11
HCF of 97 and 22 is 1.
Hence, the required number is 1.

8. A bakery sells cookies in three boxes. The three boxes contain 60, 84 and 108 number of cookies. The baker wants to sells all the cookies in any of the three boxes. The least number of cookies that he can bake, in a day, so that he is able to sell all his cookies in any of the three boxes is _______
a) 5467
b) 2243
c) 1123
d) 3780

Explanation: The three boxes contain 60, 84 and 108 number of cookies.
60 can be factorized as 2 × 2 × 3 × 5, 84 as 2 × 2 × 3 × 7 and 108 as 2 × 2 × 3 × 3 × 3
To find the least number of cookies that can be filled in the container, we have to find the LCM of the three numbers
LCM of 60, 84 and 108 = 2 × 2 × 3 × 3 × 3 × 5 × 7 = 3780
Hence, the least number of cookies that can be filled in the container is 3780.

9. Two buckets contain 546 and 764 liters of water respectively. What will be maximum capacity of container which can measure the water of either buckets exact number of times?
a) 108
b) 54
c) 34
d) 456

Explanation: The two buckets contain 546 and 764 liters of water.
546 can be factorized as 2 × 2 × 3 × 3 × 3 × 5 and 764 as 2 × 2 × 3 × 3 × 3 × 7.
To find the maximum capacity of the container which can measure the water of either buckets exact number of times, we have to find the HCF of the two numbers.
HCF of 546 and 764 = 22 × 33=108
Hence, the maximum capacity of the container is 108 liters.