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Generalizing the Problem

Generalizing is making broad statements about a solution, and realizing that the solution to one problem may apply to many other problems.

 
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Here is a simply problem to demonstrate generalization:

Let us investigate the divisibility by 2 (test of divisibility by 2).

 

Remainder

Exactly divisible or not

420 ÷ 2

0

Divisible

321 ÷ 2

1

Not Divisible

8922 ÷ 2

0

Divisible

2423 ÷ 2

1

Not Divisible

124 ÷ 2

0

Divisible

25 ÷ 2

1

Not Divisible

326 ÷ 2

0

Divisible

427 ÷ 2

1

Not Divisible

28 ÷ 2

0

Divisible

629 ÷ 2

1

Not Divisible


After observing the division facts on the above table, we may generalize that:

  • the numbers ending in 0, 2, 4, 6, 8 are divisible by 2
  • the numbers ending in 1, 3, 5, 7, 9 are not divisible by 2

Further, we may generalize that all even numbers are divisible by 2, and that all odd numbers are not divisible by 2.

This generalization is now useful in answering question such as, "Is 45974 divisible by 2?"

We don't need to do any math. We can state the answer is "yes," since 45974 an even number.

 
 
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