Prime Factor Circles - Polypad Pointer

Рет қаралды 2,427

Күн бұрын

Пікірлер: 11

@mutafyanny 4 ай бұрын

Thank you you two, mathigon, the factor circles i use, math for love, inventing the board game (even if i don't own one)

@Treborino Жыл бұрын

Firstly, thank you for the brilliant Mathigon and amazing Polipad. It is an inexhaustible resource for any school math teacher, and videos on this channel are the best source of inspiration. First hint: How to go through all the factors of a specific number without getting lost? The answer one can find in this video. You should group them in pairs, which product is the main number. Second hint: How can the number of factors be odd if all of them are grouped in pairs? The answer is a bit trickier. The main number must be a full square, so the pair for a certain factor is this number itself. Thus, the number with exactly seven factors is a full square. Third hint: How to find the number of factors without finding them? The Prime Factors Circles module on Polypad lets us decompose any number on its prime factors in a few seconds. However, finding all the factors will take more time, even if we do it in a well-organized way. This question is the most complicated, but at the same time the most promising. Let’s take the number 18, for example. 18=2*3*3. So we have one 2 and two 3s. If we want to construct a factor of 18 from these primes, how can we do this? We can take 2s or 3s or the both. How many 2s can we take? We can take 0 of it (that is not to take 2s at all) or one 2. How many 3s can we take? We can take 0 of it (that is not to take 3s at all) or one 3 or two 3s. Now let get it together. We have 2 options for 2s and 3 options for 3s. They are independent, so for any of 2 choices for 2s we can choose one of the 3 choices for 3s. Thus, the number of factors of 18 (that is, the number of different combinations of one 2 and two 3s) is 2*3=6. Indeed, if we take zero 2s and zero 3s we’ll get 1, one 2s and zero 3s - 2, zero 2s and one 3 - 3, one 2 and one 3 - 6, zero 2 and two 3s - 9, one 2 and two 3s - 18. Thus, the number of factors of a certain number is the product of exponents increased by one of all distinct prime factors of this number. Now we're approaching the main question. What number has exactly seven factors? The number 7 itself is a prime number, so it can’t be obtained by multiplying other numbers than 7 and 1. Therefore, the only way for a number to have exactly seven factors is to be a 6th power of a prime number. The smallest number of this kind is 2 in power 6, which is 64. In addition, we can take 3 in power 6 that is 729 or a 6th power of any other prime number.