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Suppose for @,b € Z,a b. Which of the following is NOT necessarily true?If @ | cthen b | c01 | aand b | 0.If b | a, then a = #b.For any integer €, @ bx...

Question

Suppose for @,b € Z,a b. Which of the following is NOT necessarily true?If @ | cthen b | c01 | aand b | 0.If b | a, then a = #b.For any integer €, @ bx

Suppose for @,b € Z,a b. Which of the following is NOT necessarily true? If @ | cthen b | c 01 | aand b | 0. If b | a, then a = #b. For any integer €, @ bx



Answers

Let $a$ and $b$ be numbers such that $-a<b+1<a$. Which of the following must be true? a. $a \leq 0$ b. $b \leq a$ c. $b.a + 1$ A. I only B. I and II C II only D. I, II, and III

So in this functional comparing a factorial times the factorial see factor or be factorial times see fact oils and parentheses. And then we're comparing this to a factorial times be fact, royal being in the parentheses this time. So it looks more like this and where I have to say all these equal or not, the answer of this question is gonna be This statement is true for the reason that it really doesn't matter the order you multiplies the same with addition and multiplication. The properties say that basically, it doesn't matter whether we multiply a factorial, times be factorial first or if we do b times C because we're gonna get the same product.

So this function we have a factorial to the power of the and then be it's our eight of the power be factual and were asked Is this statement true? So this statement is gonna be false because on the left side were first taking our base and doing the factorial of that will go a counter example after this and then applying it to the power bi. But here, B is the one that's fact wrote us basically, here we do the base factorial here we do the exponent popular. It's not a given example. Let's just say is three and b is for so on the left side. Then we're gonna have three times three fact oil, which is six to the power for, you know, And then over here we're gonna have three to the power of four factorial, which is 24. So right now we're going to see that this is not true. And that is why this statement is false. This is a counter example

So in this question, we're comparing a B factorial vs a factor will be factorial and we're off to see Is this statement true of false? So basically a V factorial needs like if let's just give and he ran and values, let's say, is five nbc'ssaturday so a B factorial is gonna be 35 factorial, whereas a factory will be factory. Lt's gonna be five factorial times seven factorial and already we conceded this is definitely gonna be false because 35 factorial is huge like almost 9 35 34 by 33. That's gonna be a very large number. Where is five factorial times seven factor unless we need relatively small. So basically the difference is that in a factorial be factual right here we do each number individually we do if I told you about factorial of that whereas in this one we multiply a times B first because it's in the parentheses, which means we get a much larger number. And so basically this is why these two are different

So is this statement true or false? A factorial plus B factorial equals B factorial plus a factorial. Well, I say it's true. And here's why. Because a factorial has to be a natural number. Natural numbers are numbers like 12345 etcetera, also known as accounting numbers and be factorial is also a natural number, and natural numbers are really numbers. And we have the community of property of real numbers that tells us that we can switch the order when we're adding and it's equivalent. Therefore, a factorial plus B factorial is equal to be factorial plus a factorial.


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