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Divisibility Rules
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Foundational skills
Recognise the meaning of divisibility
\[ \text{divisible by } 6 = \text{multiple of } 6 \]
Understand that divisibility means exact division with no remainder.
Divisibility by 10
\[ 4730 \div 10 \text{ ?} \]
Test divisibility by 10 using the last digit rule.
Divisibility by 5
\[ 38472 \div 5 \text{ ?} \]
Test divisibility by 5 using the last digit rule.
Divisibility by 2
\[ 57431 \div 2 \text{ ?} \]
Test divisibility by 2 using the last digit rule.
Basic divisibility rules
Divisibility by 3 using digit sum
\[ 48273: \text{ digit sum } = 24 \]
Test divisibility by 3 by checking if the digit sum is divisible by 3.
Divisibility by 9 using digit sum
\[ 73629: \text{ digit sum } = 27 \]
Test divisibility by 9 by checking if the digit sum is divisible by 9.
Divisibility by 4 using last two digits
\[ 35724: \text{ last two } = 24 \]
Test divisibility by 4 by checking if the last two digits form a number divisible by 4.
Divisibility by 8 using last three digits
\[ 451280: \text{ last three } = 280 \]
Test divisibility by 8 by checking if the last three digits form a number divisible by 8.
Divisibility by 6 using combined rule
\[ \div 6 = \div 2 \text{ AND } \div 3 \]
Test divisibility by 6 by checking divisibility by BOTH 2 AND 3.
Advanced divisibility rules
Divisibility by 11 using alternating sum
\[ 918390: 9-1+8-3+9-0 = 22 \]
Test divisibility by 11 using the alternating digit sum rule.
Divisibility by 12 using combined rule
\[ \div 12 = \div 3 \text{ AND } \div 4 \]
Test divisibility by 12 by checking divisibility by both 3 AND 4.
Divisibility by 15 using combined rule
\[ \div 15 = \div 3 \text{ AND } \div 5 \]
Test divisibility by 15 by checking divisibility by both 3 AND 5.
Divisibility by 18 using combined rule
\[ \div 18 = \div 2 \text{ AND } \div 9 \]
Test divisibility by 18 by checking divisibility by both 2 AND 9.
Divisibility by 20 using last two digits
\[ \text{Last two: } 00, 20, 40, 60, 80 \]
Test divisibility by 20 by checking if the last two digits form a number divisible by 20.
Divisibility by 25 using last two digits
\[ \text{Last two: } 00, 25, 50, 75 \]
Test divisibility by 25 by checking if the last two digits are 00, 25, 50, or 75.
Divisibility by 24 using combined rule
\[ \div 24 = \div 3 \text{ AND } \div 8 \]
Test divisibility by 24 by checking divisibility by both 3 AND 8.
Finding missing digits
Missing digit for divisibility by 3
\[ 4d723 \div 3: \text{ find } d \]
Find all values (0–9) of a missing digit that make a number divisible by 3.
Missing digit for divisibility by 9
\[ 72d45 \div 9: \text{ find } d \]
Find all values (0–9) of a missing digit that make a number divisible by 9.
Missing digit for divisibility by 4
\[ 385d \div 4: \text{ find } d \]
Find all values (0–9) of a missing digit in the last two positions that make a number divisible by 4.
Missing digit for divisibility by 6
\[ 5d284 \div 6: \text{ find } d \]
Find all values (0–9) of a missing digit that make a number divisible by 6.
Missing digit for divisibility by 8
\[ 46d24 \div 8: \text{ find } d \]
Find all values of a missing digit in the last three positions that make the number divisible by 8.
Missing digit for divisibility by 11
\[ 3d5742 \div 11: \text{ find } d \]
Find all values (0–9) of a missing digit that make a number divisible by 11.
Testing multiple divisibility
Identify all small divisors
\[ 2340: \text{ which of } 2,3,4… \text{ ?} \]
Test a number against multiple divisibility rules.
Evaluate compound divisibility statements
\[ \div 4 \text{ but not } \div 8 \text{ ?} \]
Evaluate compound statements about divisibility.
Find numbers meeting multiple conditions
\[ \text{Which is } \div 3 \text{ AND } \div 4 \text{ ?} \]
Identify which numbers in a list satisfy multiple divisibility conditions.
Applications
Check if a number has a given factor
\[ \text{Is } 6 \text{ a factor of } 72546 \text{ ?} \]
Use divisibility rules to identify factors without performing long division.
Determine if a fraction gives a whole number
\[ \frac{73413}{9} = \text{whole?} \]
Use divisibility to determine if a division yields a whole number.
Find largest divisor from a set
\[ \text{Largest from } \{2,3,4,5,6,8,9\} \]
Identify the largest divisor of a number from a given set.
Special cases
Divisibility and remainders
\[ 23471 \div 9 = \text{? remainder} \]
Use digit sum to find remainder on division by 9.
Divisibility patterns in products
\[ 24 \times 35 \times 11 \div 8 \text{ ?} \]
Determine divisibility of a product without calculating it.
Always/sometimes/never divisibility
\[ 6n \div 3 \text{ : always?} \]
Determine whether an expression is always, sometimes, or never divisible by a given number.
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