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Divisibility Rules can be the basis for some very interesting mathematics. Unfortunately, these “rules” are often taught by rote without understanding. In such cases the mathematics has been left out! DON’T leave out the mathematics! The purpose of this activity is to help you understand both informally and formally why the divisibility rules work. Understanding = Power!
Divisibility Rule for 5:
State the divisibility rule for five: Picture Proof for 3 digit positive integer: Can you explain why it works?
“Formal” Proof for 3 digit positive integer using the divisibility theorems Any 3 digit positive integer can be written in the form a(100)+b(10)+c, a, b, c Z+ Our Goal to show that if 5|c then 5| (a(100)+b(10)+c) We see that 5|a(100) WHY^1? and 5|b(10) WHY? Now, by applying the theorem “If a|b and a|c then a|(a+c)“ we have if 5|c , then five divides the number. Can you explain why? (^1) You will need to use the definition of divisibility as well as the concept of closure
Divisibility Rule for 4:
A) Use base 10 blocks to discover/justify a divisibility rule for four. State the Rule: B) Use base 10 blocks to justify your rule for three digit numbers. C) Test the rule with several large numbers (greater than 1000). Explain your results. Will the rule always work? Can you use base 10 blocks to explain why? D) Can you find more than one divisibility rule for four?
Yet More Divisibility Rules
Can you state and justify the divisibility rules for 9 and 10?
