Discussion
If the remainder is 1 when m is divided by 2 and the remainder is 3 when n is divided by 4, which of the following must be true?
*This question is included in Nova Math - Problem Set F: Number Theory, question #1
| (A) | m is even. |
| (B) | ... |
| (C) | ... |
| (D) | ... |
| (E) | ... |
| (F) | ... |
The solution is
Posted: 02/10/2013 13:49
I'm looking at method one and would like an explanation as to why when an equation is a multiple of two it is known to be even? Conversely does this hold true if the multiple is 3 than the equation is odd? Or is it simply that when an equation can be simplified to a multiple of an even number it is even or an odd number it is odd?
Posted: 02/10/2013 20:16
Hi Gaine,
By definition, a number or expression is even if it is a multiple of 2. For example, 6 is even because it can be written as 2(3), and 8x + 10 is even because it can be written as 2(4x + 5).
A number or expression is odd if it can be written as 1 more than a multiple of 2, i.e., in the form 2x + 1. For example, 7 is odd because it can be written as 2(3) + 1, and 4x + 6k + 1 is odd because it can be written as 2(2x + 3k) + 1. For simplicity, we can let q equal 2x + 3k in the expression 2(2x + 3k) + 1, reducing it to 2q + 1, which matches the definition of an odd number.
Being a multiple of 3 does not make a number odd or even. We just saw that 6 is a multiple of 3, but it is even. And 9 [= 3(3)] is a multiple of 3 and it is odd: 9 = 2(4) + 1.
Nova Press
By definition, a number or expression is even if it is a multiple of 2. For example, 6 is even because it can be written as 2(3), and 8x + 10 is even because it can be written as 2(4x + 5).
A number or expression is odd if it can be written as 1 more than a multiple of 2, i.e., in the form 2x + 1. For example, 7 is odd because it can be written as 2(3) + 1, and 4x + 6k + 1 is odd because it can be written as 2(2x + 3k) + 1. For simplicity, we can let q equal 2x + 3k in the expression 2(2x + 3k) + 1, reducing it to 2q + 1, which matches the definition of an odd number.
Being a multiple of 3 does not make a number odd or even. We just saw that 6 is a multiple of 3, but it is even. And 9 [= 3(3)] is a multiple of 3 and it is odd: 9 = 2(4) + 1.
Nova Press