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Factors and Multiples - Divisibility Rules for 2, 3, 5, and 10

Grade 4ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Divisibility means that when one number is divided by another, the result is a whole number with a remainder of 00. Imagine a group of 1212 marbles being shared equally among 33 friends; each gets exactly 44 marbles with nothing left over, so 1212 is divisible by 33.

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A number is divisible by 22 if it is an even number, meaning its last digit (in the ones place) is 0,2,4,6,0, 2, 4, 6, or 88. Visualise a number line where you jump by 22 units starting from 00; you will always land on an even digit.

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A number is divisible by 33 if the sum of all its individual digits is a multiple of 33. For example, for the number 123123, you add 1+2+3=61 + 2 + 3 = 6. Since 66 is divisible by 33, 123123 is also divisible by 33. Think of breaking a number into its digits and stacking them like blocks to see if the total height matches the 33 times table.

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A number is divisible by 55 if its last digit is either 00 or 55. Picture a clock face marked in 55-minute intervals: 5,10,15,20...5, 10, 15, 20... every number ends in either 55 or 00.

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A number is divisible by 1010 if it ends with the digit 00. Imagine a stack of 1010-rupee notes; no matter how many notes you have, the total value will always end in a 00, such as 10,20,150,10, 20, 150, or 10001000.

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Factors are the numbers you multiply together to get another number. For example, 22 and 55 are factors of 1010 because 2×5=102 \times 5 = 10. If a number is divisible by 1010, it must also be divisible by its factors 22 and 55.

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Multiples are the products found in a number's multiplication table. A number is divisible by its factors. For instance, 1515 is a multiple of 55, so 1515 is divisible by 55. Visualize multiples as a repeating pattern or a skip-counting sequence on a grid.

📐Formulae

Divisibility by 2  ⟹  Last digit∈{0,2,4,6,8}\text{Divisibility by 2} \implies \text{Last digit} \in \{0, 2, 4, 6, 8\}

Divisibility by 3  ⟹  (d1+d2+d3+...)÷3 has remainder 0\text{Divisibility by 3} \implies (d_1 + d_2 + d_3 + ...) \div 3 \text{ has remainder } 0

Divisibility by 5  ⟹  Last digit∈{0,5}\text{Divisibility by 5} \implies \text{Last digit} \in \{0, 5\}

Divisibility by 10  ⟹  Last digit=0\text{Divisibility by 10} \implies \text{Last digit} = 0

Dividend=Divisor×Quotient+Remainder\text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder}

💡Examples

Problem 1:

Check if the number 276276 is divisible by 33.

Solution:

Step 1: Identify the digits of the number 276276, which are 2,7,2, 7, and 66. Step 2: Calculate the sum of the digits: 2+7+6=152 + 7 + 6 = 15. Step 3: Check if the sum 1515 is divisible by 33. Since 15÷3=515 \div 3 = 5 with a remainder of 00, 1515 is divisible by 33. Conclusion: Therefore, 276276 is divisible by 33.

Explanation:

To test for divisibility by 33, we sum the digits. If that sum is a multiple of 33, the entire original number is divisible by 33.

Problem 2:

Is the number 450450 divisible by 2,5,2, 5, and 1010?

Solution:

Step 1: Check the last digit of 450450, which is 00. Step 2: Rule for 22: Since the last digit is 00 (an even number), it is divisible by 22. Step 3: Rule for 55: Since the last digit is 00, it satisfies the rule for 55. Step 4: Rule for 1010: Since the last digit is 00, it is divisible by 1010. Conclusion: Yes, 450450 is divisible by 2,5,2, 5, and 1010.

Explanation:

Numbers ending in 00 satisfy the divisibility rules for 2,5,2, 5, and 1010 simultaneously because 1010 is a multiple of both 22 and 55.