Playing with Numbers: NCERT Class 6 Maths Exercise 3.3

Welcome, young mathematicians! In CBSE Class 6 Maths Chapter 3, Exercise 3.3 focuses on one of the most exciting tools in arithmetic: Divisibility Tests. Instead of performing tedious long division, we can use simple shortcut rules to check if a large number is divisible by 2, 3, 4, 5, 6, 8, 9, 10, or 11. Learning the concepts behind playing with numbers ex 3 3 class 6 ncert helps us solve complex fraction reductions, find prime factors, and speed up our mental math calculations. In this interactive guide by your YoLearn AI Tutor, we will break down each divisibility test with step-by-step place-value thinking, practice solved examples together, and address common exam pitfalls so you can ace this exercise easily!

Understanding the Logic Behind Divisibility Rules

Why do we use divisibility tests? Divisibility tests are shortcuts that let us inspect a number's digits to decide if it divides perfectly by a target divisor without leaving a remainder. For example, why is a number divisible by 2 if it ends in 0, 2, 4, 6, or 8? Because any multi-digit number can be split using place value. Consider 548 as 500 + 40 + 8. Since 100 is divisible by 2, any multiple of 100 is also divisible by 2. Since 10 is divisible by 2, any multiple of 10 is divisible by 2. Thus, the divisibility depends entirely on the ones place digit, which is 8. This place-value logic underpins all divisibility rules. By mastering these rules, you will quickly identify factors and multiples, making fraction calculations a breeze!

Quick Summary of Divisibility Rules

Divisibility by 2
A number is divisible by 2 if its units (ones) digit is 0, 2, 4, 6, or 8 (an even number).
Divisibility by 3
A number is divisible by 3 if the sum of all its digits is a multiple of 3.
Divisibility by 4
A number is divisible by 4 if the number formed by its last two digits (tens and ones places) is divisible by 4.
Divisibility by 5
A number is divisible by 5 if its ones digit is either 0 or 5.
Divisibility by 6
A number is divisible by 6 if it is divisible by both 2 and 3.
Divisibility by 8
A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
Divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9.
Divisibility by 10
A number is divisible by 10 if its ones digit is 0.

How to Test Divisibility by 11 Step-by-Step

  1. Step 1: Identify positions from the right — Write down the number and mark the positions of digits starting from the rightmost digit as position 1 (odd), position 2 (even), position 3 (odd), etc.
  2. Step 2: Sum the digits at odd positions — Add all the digits located at odd-numbered positions (1st, 3rd, 5th, etc. from the right).
  3. Step 3: Sum the digits at even positions — Add all the digits located at even-numbered positions (2nd, 4th, 6th, etc. from the right).
  4. Step 4: Find the difference — Subtract the smaller sum from the larger sum to find their difference.
  5. Step 5: Check the final value — If the difference is either 0 or a multiple of 11 (like 11, 22, 33), the original number is divisible by 11.

Important Exam Trap: Divisibility of 11

A common mistake students make is counting positions starting from the left instead of the right when applying the divisibility rule for 11. Remember: always label your positions from right to left (starting with position 1 on the ones place). To stay safe and score full marks, strictly follow the NCERT guideline of counting from the right! Also, do not confuse the rule of 3 (sum of digits) with the rule of 4 (last two digits).

Practice Questions with Solutions

  • Q: Test whether the number 572 is divisible by 4 and by 8. A: Step 1: To test divisibility by 4, look at the last two digits of 572, which is 72. Divide 72 by 4: 72 ÷ 4 = 18. Since 72 is completely divisible by 4, 572 is divisible by 4. Step 2: To test divisibility by 8, look at the last three digits of 572, which is 572 itself. Divide 572 by 8: 572 ÷ 8 = 71 with a remainder of 4. Since there is a remainder, 572 is not divisible by 8. Final answer: 572 is divisible by 4 but not divisible by 8.
  • Q: Using divisibility tests, determine if 1258 is divisible by 6. A: Step 1: Check divisibility by 2. The ones digit of 1258 is 8, which is an even number. So, 1258 is divisible by 2. Step 2: Check divisibility by 3. Calculate the sum of the digits: 1 + 2 + 5 + 8 = 16. Since 16 is not divisible by 3, 1258 is not divisible by 3. Step 3: A number is divisible by 6 only if it is divisible by both 2 and 3. Since 1258 is divisible by 2 but not by 3, it is not divisible by 6. Final answer: 1258 is not divisible by 6.
  • Q: Check if the number 10824 is divisible by 11. A: Step 1: Label the digits from right to left: 4 (1st - odd), 2 (2nd - even), 8 (3rd - odd), 0 (4th - even), 1 (5th - odd). Step 2: Find the sum of digits at odd places: 4 + 8 + 1 = 13. Step 3: Find the sum of digits at even places: 2 + 0 = 2. Step 4: Find the difference of these sums: 13 - 2 = 11. Step 5: Check if the difference is divisible by 11. Since 11 is divisible by 11, the original number is divisible by 11. Final answer: Yes, 10824 is divisible by 11.
  • Q: Find the smallest digit that should replace the blank space in _6724 to make the number divisible by 3. A: Step 1: Let the missing digit be x. The number is x6724. First find the sum of the known digits: 6 + 7 + 2 + 4 = 19. Step 2: The total sum of digits including x must be a multiple of 3. So, the sum is 19 + x. Step 3: Find the smallest single-digit number x such that 19 + x is divisible by 3. The multiples of 3 near 19 are 21, 24, etc. For 19 + x = 21, we get x = 2. Final answer: The smallest digit to replace the blank space is 2.

Frequently Asked Questions

What is the easiest way to check if a number is divisible by 6?

To check if a number is divisible by 6, simply apply the divisibility rules of both 2 and 3. If the number ends in an even digit and the sum of its digits is a multiple of 3, then it is divisible by 6.

How is the divisibility rule of 4 different from the rule of 8?

The divisibility rule of 4 requires you to check only the last two digits of a number. On the other hand, the rule of 8 requires checking the last three digits of the number.

If a number is divisible by 9, is it always divisible by 3?

Yes, because if a sum of digits is divisible by 9, it is automatically divisible by 3 since 9 is a multiple of 3.