I have this question to work on, and I don't know where to start investigating the answer:
The product of any two (whole) numbers each of which leave a remainder of 1 on dividing by 7, also leaves a remainder of 1 on dividing by 7. Why?
I THINK that I can see a quadratic in there ( (n+1)(2n+1) ); and when I multiply any variation out, there's always a remainder 1.
Can anyone confirm the link; and point me where to go next? Could i use a diagram to explain it? Thanks.
Charlie
An integer that leaves a remainder of 1 when divided by 7 can be represented by
7m+1, or 7n+1, where m, n are integers.
The product is thus:
(7m+1)(7n+1)
Expand the product and complete the proof.
you will have to exolain it by yourself😿💦💦💦💦
To investigate the question, you can start by examining the given condition: "The product of any two (whole) numbers each of which leave a remainder of 1 on dividing by 7 also leaves a remainder of 1 on dividing by 7."
Let's analyze this condition step by step.
First, let's consider two numbers, let's call them n and m, that both leave a remainder of 1 when divided by 7. In other words, we can write them as:
n = 7a + 1
m = 7b + 1
Where a and b are integers.
Now, let's calculate the product of n and m:
n * m = (7a + 1) * (7b + 1)
Using simple algebra, we can expand this expression:
n * m = 49ab + 7a + 7b + 1
Now, let's observe the remainder when this product is divided by 7.
n * m = 49ab + 7a + 7b + 1 = 7(7ab + a + b) + 1
As you can see, the expression n * m can be written as 7 times some integer plus 1. Therefore, the remainder of n * m when divided by 7 is indeed 1.
To summarize, when you multiply any two whole numbers that leave a remainder of 1 when divided by 7, the product will also leave a remainder of 1 when divided by 7.
Regarding your idea of using a quadratic expression to explain this, you are correct. The expression you mentioned, (n + 1)(2n + 1), represents the product of two whole numbers, each leaving a remainder of 1 when divided by 7. By expanding this quadratic expression, you will arrive at the same result: the product leaves a remainder of 1 when divided by 7.
If you want to provide a diagram to explain this concept, you can create a visual representation of the process of dividing integers by 7 and how remainders affect the final product. For example, you can use a diagram to illustrate the step-by-step calculations for multiplying two numbers that leave a remainder of 1 when divided by 7. This could help demonstrate the consistency of the remainder of 1 in the final product while reinforcing the concept.