Question 1

What is arithmetic in mathematics?

Accepted Answer

The branch of mathematics dealing with numbers and basic operations like addition, subtraction, multiplication, and division — Arithmetic is a branch of mathematics that focuses on numbers and the basic operations performed on them, such as addition, subtraction, multiplication, and division. It forms the foundation for many other mathematical topics.

Question 2

Which of the following is NOT a property of arithmetic operations?

Accepted Answer

Reflective property — The main properties of arithmetic operations are commutative, associative, distributive, identity, and inverse properties. Reflective property is not a property of arithmetic operations.

Question 3

Which property of addition states that changing the order of numbers does not change the sum?

Accepted Answer

Commutative property — The commutative property of addition states that a + b = b + a, meaning the order of numbers can be changed without affecting the sum.

Question 4

Explain the associative property of multiplication with an example.

Accepted Answer

The associative property of multiplication states that the way numbers are grouped does not affect the product. For example, (2 × 3) × 4 = 2 × (3 × 4). Both equal 24. — The associative property means that when multiplying three or more numbers, the grouping of numbers does not change the result. For instance, (2 × 3) × 4 = 6 × 4 = 24, and 2 × (3 × 4) = 2 × 12 = 24, so both sides are equal.

Question 5

State and explain the distributive property of multiplication over addition.

Accepted Answer

The distributive property states that multiplying a number by a sum is the same as multiplying the number by each addend and then adding the results. For example, a × (b + c) = a × b + a × c. — The distributive property connects multiplication and addition. It shows that to multiply a number by a sum, you can multiply the number separately by each addend and then add the products. For instance, 3 × (4 + 5) = 3 × 4 + 3 × 5 = 12 + 15 = 27.

Question 6

Which number is the identity element for multiplication?

Accepted Answer

1 — The identity element for multiplication is 1 because any number multiplied by 1 remains unchanged, e.g., a × 1 = a.

Question 7

Verify the commutative property of addition using 3 beads and 5 beads.

Accepted Answer

The commutative property of addition states that changing the order of numbers does not change the sum. For example, 3 beads + 5 beads = 8 beads and 5 beads + 3 beads = 8 beads. Both sums are equal, verifying the property. — By counting beads arranged first as 3 and then 5, and then reversing the order to 5 and then 3, the total remains 8 in both cases. This shows that addition is commutative.

Question 8

Define a factor of a number and give an example.

Accepted Answer

A factor of a number is a number that divides it exactly without leaving a remainder. For example, 3 is a factor of 12 because 12 ÷ 3 = 4 with no remainder. — Factors are numbers that divide another number completely. For instance, 1, 2, 3, 4, 6, and 12 are factors of 12 because each divides 12 exactly.

Question 9

What are multiples of a number? Provide an example with the number 5.

Accepted Answer

Multiples of a number are obtained by multiplying the number by integers. For example, multiples of 5 are 5, 10, 15, 20, and so on. — Multiples are the products of a number and any integer. For 5, multiplying by 1, 2, 3, 4 gives 5, 10, 15, 20 respectively, which are its multiples.

Question 10

Differentiate between prime and composite numbers with examples.

Accepted Answer

A prime number has exactly two factors: 1 and itself, such as 2, 3, 5, and 7. A composite number has more than two factors, like 4, 6, 8, and 9. — Prime numbers cannot be divided exactly by any number other than 1 and themselves. Composite numbers have additional factors. For example, 5 is prime because only 1 and 5 divide it, while 6 is composite because it has factors 1, 2, 3, and 6.

Question 11

Find all factors of 18 using division.

Accepted Answer

Factors of 18 are numbers that divide 18 exactly: 1, 2, 3, 6, 9, and 18. — Dividing 18 by numbers 1 through 18, the ones with zero remainder are factors. 18 ÷ 1 = 18, 18 ÷ 2 = 9, 18 ÷ 3 = 6, 18 ÷ 6 = 3, 18 ÷ 9 = 2, 18 ÷ 18 = 1.

Question 12

Which of the following numbers is divisible by 3?

Accepted Answer

135 — A number is divisible by 3 if the sum of its digits is divisible by 3. For 135, the sum is 1 + 3 + 5 = 9, which is divisible by 3, so 135 is divisible by 3.

Question 13

What is the divisibility rule for 5?

Accepted Answer

A number is divisible by 5 if its last digit is 0 or 5 — A number ending with 0 or 5 can be divided exactly by 5 without remainder. For example, 25 and 40 are divisible by 5.

Question 14

Is the number 124 divisible by 4? Justify your answer.

Accepted Answer

False. 124 is not divisible by 4 because the last two digits, 24, are divisible by 4 but 124 ÷ 4 = 31 with no remainder, so the statement is actually True. — The divisibility rule for 4 states that if the last two digits of a number are divisible by 4, then the whole number is divisible by 4. Since 24 is divisible by 4, 124 is divisible by 4.

Question 15

Find the prime factorization of 36 using a factor tree.

Accepted Answer

The prime factorization of 36 is 2 × 2 × 3 × 3 or 2² × 3². — Start dividing 36 by the smallest prime 2: 36 ÷ 2 = 18; 18 ÷ 2 = 9; 9 ÷ 3 = 3; 3 ÷ 3 = 1. The prime factors are 2, 2, 3, and 3.

Arithmetic

Arithmetic — Study Notes

Introduction

Properties of Operations

Factors and Multiples

Practice Questions — Arithmetic

All 8 Chapters in Ganita Prakash