6

For high school level, showing non-primality is fair game, provided simple divisibility criteria can resolve it. Showing primality for a number such as the one you posted is definitely not fair game. One guideline would be: If the teacher can't show it easily, then it's not appropriate for students.


4

I would recommend searching for a good textbook to work from. A quick Google search on multivariable calculus brought up Single and Multivariable Calculus Early Transcendentals by David Guichard. This textbook appears to introduce the reader to the basics in the introductory chapters, and then goes on to explore the more advanced material. I suspect that ...


4

It is rather redundant to first study differential and integral calculus, followed by AP Calculus BC, only to then return to calculus 1 and 2. These courses are essentially equivalent to the first three and unless you would like to review there is not much point in studying them again. Your plan otherwise seems fine, although knowing the context of why you ...


4

By the axiomatic definition of the natural numbers, $2$ is the successor of the successor of $0$, and $5$ is the successor of the successor of the successor of the successor of the successor of $0$. Then using the axioms, $2+2=5$ can be proven to be false, full stop. Any attempt that you would make will involve other axiomatisations or other definitions of ...


3

How to make sense of that rule for divisibility by $4$: it's not saying to add the last two digits; it's merely saying to look at the last two digits. Because $4$ divides $100$, a number is divisible by $4$ if and only if its last two digits (ten's place and one's place) are divisible by $4$. Robert Israel's answer gives a method for determining whether a ...


3

I would describe mathematical maturity as being like muscle-memory for sports. When I play squash there are certain things that are so ingrained it takes thought and effort to do them wrong: when I serve I know where my racquet is and where the ball is to the extent that I don't look at either of them, I look at where I want to put the ball. Mathematical ...


3

I recently finished a course in commutative algebra, and let me tell you I completely get it. Although I don't really know what you're specifically struggling with let me tell you a little about the problems I faced and the things I did to remedy it after I got completely demolished on the final. I took commutative algebra in my third year of undergraduate ...


2

$y''+ xy = 0$ is a homogeneous linear differential equation of order $2$. The set of all solutions if this equation is a two-dimensional real vector space. Consequence: $y''+ xy = 0$ does not have a unique solution.


2

You can get infinitely many solutions of the type $\sum a_nx^{n}$. You get the relation $a_{n+3}=\frac {a_n} {(n+2)(n+3)}$ with the condition $a_2=0$. You can assign any value for $a_0$ and $a_1$ and find $(a_n)$. The series necessarily converges in a neighborhood of $0$.


2

You get confused by viewing the evaluation of an expression as a matter of gradually replacing symbols in a formula by other symbols. Doing so is extremely convenient and a great timesaver, but only as long as you know where you need to insert parentheses in the right places such that you won't get nonsense out of it. And the way to know that is to know that ...


2

When you say that filling in the variable $x=-2$ and $y=5$ results in: -5*-2²-(-2+5) (fill in variables) you are changing the unary operator '-' (that takes a number $x$ and outputs $-x$ ... this is what the '-' does to the '2' in '-2') into a binary operator that subtracts a number from another number. That is really the mistake you make here. So ...


2

The number is a prime. This cannot be found out in reasonable time without a computer.


1

$$ 2 + 2 = 5 \quad\text{(mod $1$)} $$


1

Tens place even and units $0$, $4$ or $8$ (i.e. divisible by $4$), or tens place odd and units $2$ or $6$ (even but not divisible by $4$).


1

Insightful question: As one can see by the comments, this is a matter of opinion. As a textbook author in 'this modern age', I have given this issue some thought. So here is my opinion. It is true that many routine problems involving normal distributions are easily solved using software. To an extent that statistics and probability texts of the future will ...


1

Hint: The triangles formed by joining the vertices of the squares as shown in the figure are similar.


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