Find the Rule Practice Test Worksheets
Build confidence with printable Find the Rule worksheets and focused number reasoning practice tests. Students learn to test factors, multiples, primes, powers, digit properties, and number patterns while improving accuracy with logical elimination questions.
For students who can calculate fluently, but need practice proving which rule fits every number in a set.
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What Do Find the Rule Questions Look Like?
Find the rule questions present a set of numbers and ask you to determine the mathematical property that every number in the set shares. Here is a worked example.
Check Every Number
The numbers are alike in some ways.
Select ONE of the options to say one way in which they are alike.
18 108 153 180
Options:
A) Their tens digit is greater than their units digit.
B) They are all multiples of 9.
C) They are all three-digit numbers.
D) They are all even numbers.
E) They are all multiples of 4.
Worked Method
Check A: 18 and 108 do not have a tens digit greater than their units digit, so A fails.
Check B: 18 ÷ 9 = 2, 108 ÷ 9 = 12, 153 ÷ 9 = 17, and 180 ÷ 9 = 20. All four work.
Check C: 18 is a two-digit number, so C fails.
Check D: 153 is odd, so D fails.
Check E: 18 and 153 are not multiples of 4, so E fails.
Answer: B, they are all multiples of 9.
Tip: Do not stop when one option sounds plausible. A single counterexample eliminates a rule.
Rules to Help You Solve Find the Rule Questions
To solve find the rule questions like the example above, test candidate properties against all numbers. Follow these four rules:
Rule 1: Check every number
The correct option must fit every number in the set. One counterexample is enough to eliminate an option.
Example: In the set 18, 108, 153, and 180, three numbers are even, but 153 is odd. The rule “They are all even numbers” therefore fails.
Rule 2: Separate factors from multiples
A factor divides a target number exactly, with no remainder. A multiple is the result of multiplying a number by a whole number.
Example: 8 is a factor of 72 because 72 ÷ 8 = 9. In the other direction, 72 is a multiple of 8 because 8 × 9 = 72.
Rule 3: Recognise special number patterns
Squares have the form n × n, and cubes have the form n × n × n, where n is a whole number. Triangular numbers are sums of consecutive whole numbers starting at 1: 1, 3, 6, 10, 15, and so on.
To test “1 less than a square”, add 1 and check for a square. To test “1 more than a square”, subtract 1 instead.
Example: 8, 143, and 323 are each 1 less than a square: adding 1 gives 9 = 3², 144 = 12², and 324 = 18².
Rule 4: Use digit tests carefully
Digit-sum, palindrome, and tens-units rules can look convincing, but one counterexample is enough.
Example: 484, 676, 707, and 808 are palindromes because each reads the same forwards and backwards.
Two More Worked Examples
Here are two more examples showing how to distinguish factors from multiples and avoid partial-match traps.
Distinguish Factors from Multiples
The numbers are alike in some ways.
Select ONE of the options to say one way in which they are alike.
3 20 120
Options:
A) They are all factors of 60.
B) They are all factors of 120.
C) They are all multiples of 12.
D) They are all multiples of 5.
E) They are all 1 less than a square number.
Worked Method
Check A: 120 is not a factor of 60, so A fails.
Check B: 120 ÷ 3 = 40, 120 ÷ 20 = 6, and 120 ÷ 120 = 1. All three are factors of 120.
Check C: 3 and 20 are not multiples of 12, so C fails.
Check D: 3 is not a multiple of 5, so D fails.
Check E: Add 1 to 20 to get 21. This lies between 4² = 16 and 5² = 25, so it is not a square. Therefore, 20 is not 1 less than a square number, and E fails.
Answer: B, they are all factors of 120.
Tip: Divide the target number by each number in the set. Every result must be a whole number.
Avoid a Partial Match
The numbers are alike in some ways.
Select ONE of the options to say one way in which they are alike.
16 66 121
Options:
A) They are all 1 more than a multiple of 5.
B) They are all multiples of 6.
C) They are all even numbers.
D) They are all palindromic numbers.
E) They are all square numbers.
Worked Method
Check A: Subtract 1 from each number: 15, 65, and 120. These are all multiples of 5, so A works.
Check B: 16 and 121 are not multiples of 6, so B fails.
Check C: 121 is odd, so C fails.
Check D: 16 is not a palindromic number, so D fails.
Check E: 16 = 4² and 121 = 11², which makes this option tempting. However, 66 lies between 8² = 64 and 9² = 81, so it is not a square. E fails.
Answer: A, they are all 1 more than a multiple of 5.
Tip: The trap is choosing a familiar property that fits only part of the set. Spotting two squares is not enough: the rule must also fit 66.