[Intl] Problem Sum No. 4: Prime Factorization

What prime numbers have the product of 111111?

  • Asked by Rebecca C
  • Date: 21 March 2013
  • Source: International School; Gifted Math Program, Grade 5

Variety is the spice of life. Without different things to do, we may just die of boredom. So it is the same with Mathematics — explore and learn about different Maths topics to keep interest in the subject alive. That’s my personal but serious opinion; tell me if you agree or disagree in the “Leave a Reply” comment box at the end of this article.

So far, we’ve looked at a few problem sums for Primary 2 Maths, using both Comparison Model and Part-Whole Model to solve them. Like this question about two persons’ weights and another on dividing sweets amongst children.

For a change, let’s look at a problem sum from an International School. Recently, my friend, Rebecca, asked me the Maths question above. It’s about finding the prime factors of a certain number. And how I used a “special trick” to solve it quick…

Numbers and Their Prime Factors

What are the prime factors of a number? Simply put, every whole number greater than 1 can be expressed as a product of other numbers — these are its factors — all of which are prime numbers. Some examples:

  • 4 = 2 x 2. Prime factor: 2 (twice)
  • 6 = 2 x 3. Prime factors: 2 and 3
  • 12 = 2 x 2 x 3. Prime factors: 2 (twice) and 3
  • 60 = 2 x 2 x 3 x 5. Prime factors: 2 (twice), 3 and 5
  • 231 = 3 x 7 x 11. Prime factors: 3, 7 and 11

Obviously, every prime number has itself as its prime factor and nothing else, since 1 doesn’t count as prime. That is, although 5 = 5 x 1, the prime factor of 5 is just 5. So the prime factor for a prime number P is simply P itself. To learn more, check Wikipedia for other details about prime factors.

The process of finding the prime factors of a number is referred to as integer or prime factorization. And although it is a fact that every positive integer (greater than 1) has a unique prime factorization, there is no guarantee that its prime factors can be found easily. Of course, there is help online, like this prime factor calculator.

Tests of Divisibility — A Neat Trick

Looking for the prime factors of a certain number is actually finding prime numbers that can divide into that number exactly, starting with the smallest prime number (i.e. 2). This crude trial-division method is the simplest solution to factorize any number, while sophisticated algorithms are used in the field of cryptography, for example.

As division is needed, tests of divisibility come in useful to decide which prime numbers to test as possible factors. Many of us know at least one or two of these tests, even though we might not recognize them as such. We know that a number is divisible by 5 when its ending digit is a 5 or 0. We also know that even numbers are divisible by 2.

How do we know if a number can divide by 3? Simple, just sum its digits and see if this sum divides by 3 exactly. If needed, repeat this summation step with the current sum to get a smaller number to make things easy. Let’s try it out:

  • Is 3578916 divisible by 3?
     
    3 + 5 + 7 + 8 + 9 + 1 + 6 = 39
    (optional step) 3 + 9 = 12
    (optional step) 1 + 2 = 3
     
    As 3 (or 12, 39 from earlier steps) is divisible by 3, so 3578916 is divisible by 3.

The other divisibility tests are harder to remember, but no harm taking a look as a way to explore Maths. I found this website which teaches you about “Divisibility by prime numbers under 50”.

It Pays To Be Observant

Even though Problem Sum #4 isn’t about using Models, the Analysis step you have now learned can still be applied. Basically, we need to be clear what the question is asking and know what steps we can intelligently skip by being observant.

The most obvious fact is the number in the Maths question is odd, so 2 can’t be one of its prime factor. But there is one more, which helped to “reduce” the question so that we can deal with a smaller number. And tests of divisibility are selectively applied to quickly get the answer. Such tricks are explained in the worked example.

Sometimes, logical steps are all that are required to solve a Maths problem. Look first, then apply the correct methods.

Don’t over-complicate things; simple can be better.

Answer Sheet

Click below to reveal the Answer Sheet. You can also download the PDF file for offline reading and printing.

Click here to open and view the Answer Sheet

Click hereto get the Answer Sheet PDF file. To save it, right-click for options.

The Answer Sheet is annotated in a specific but consistent way, as I’ve explained here.

Let Me Know What You Think

Problem Sum #4 is probably not common in your regular diet of local Maths problem sums. Let me know if this article has been useful and whether you want me to cover more questions like this. Perhaps it has sparked some interest in exploring Mathematics or helped you see things differently. Drop me a comment or two by replying below.

As the Answer Sheet to this International School-standard Maths question illustrates, the student can approach a Maths problem in a logical and intelligent way, so that there is no struggle in finding the answer. This problem sum is now part of our No Problem Sums Library for easy lookup.

If you come across Maths questions of similar nature, send them to No Problem Sums so that I can turn them into worked examples and add their solutions to the Library as well.

Click hereto send me your problem sums. I’ll have a go at solving them soon…

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