What is the sum of the first 500 counting numbers...i.e. 1+2+3+4+5.......+496+497+498+4...
This is not as difficult as it sounds. You just need to break the problem into something easier to solve. In this case, you can consider the numbers in pairs. 1+500=501, and 2+499 is also 501. If you pair up all the numbers, you can go all the way to the middle where 250 + 251 = 501. You know that this is the middle because the numbers are just one jump apart, and the first number tells you how many pairs there are. So the total is 250 pairs times the total for each pair: 250 x 501 = 125250 The same process works for your second problem. 1+499 = 500. 249+251 = 500. In this case we're going by twos, so you don't have 249 pairs, you have half as many... or 125 pairs (yah, half of 249 is 124.5, but don't forget you're starting from 1, not 0). Half as many pairs makes sense because you're counting by twos! 125 x 500 = 62500 Hope that helps!
How many unique pairs of integers can be multiplied to get 160?
Take the prime factorization of your number. It will be expressed in the form[math]x_1^ax_2^b...x_n^z[/math].The number of factors of a number can thus be expressed as [math](a+1)(b+1)...(z+1)[/math]. In your case this is [math](5+1)(1+1)=12[/math]. As each factor must have a pair, this can be seen as the number of positive factorizations by dividing by two, as long as the number is not a perfect square, in which case one of the pairs is itself. In this case we should take the number divided by two and round up. It can be seen from this that all odd numbers generated here should be perfect squares. We can see that the general formula is two times the ceiling of [math]x_1^ax_2^b...x_n^z[/math] divided by two.Looking at the factorizations that others have listed below should give you intuition as to why. A prime number raised to a power x has x+1 factors.
Let x and y be odd primes. How many sets of two numbers multiply to 32*x*y, and have an even sum?
Alright, so let’s say we have two numbers [math]p[/math] and [math]q[/math] so that:[math]pq = 32xy[/math]Firstly, since [math]x[/math] and [math]y[/math] are prime, and cannot be “broken down”. We realize that we must either make both of them be in [math]p[/math], both of them in [math]q[/math], [math]x[/math] in p and [math]y[/math] in [math]q[/math] or [math]x[/math] in q and [math]y[/math] in p.Now for each of these four possibilities, we need to look at the various ways of splitting up 32 amongst [math]p[/math] and [math]q[/math]. 32 is prime factored as [math]2^5[/math]. Hence we need to distribute 5 2s amongst [math]p[/math] and [math]q[/math]. There are normally 6 ways to do this, [math]5[/math] to [math]p[/math], 0 to [math]q[/math], or [math]4[/math] to [math]p[/math], 1 to [math]q[/math] and so on. But in your case, there are only 4 ways to do this, since we must give atleast one two to each of [math]p[/math] and [math]q[/math] to make sure they are both even, so that we have an even sum. (The only other way we can have an even sum is if we make them both odd. Do you see why this is not possible?)So, since there are 4 ways to distribute [math]x[/math] and [math]y[/math], and 4 ways to distribute the 2s, we have a total of 16 ways. If you don’t mind the order (eg: [math]p[/math] = 10, [math]q[/math]= 5 is the same as [math]q[/math]=10, [math]p[/math] = 5), then you can divide it by two, to get 8 ways of doing so.In general, you can notice a cool pattern. We have three prime factors, 2, [math]x[/math] and [math]y[/math], occurring 5, 1 and 1 times respectively. So the total number of ways to create a product of two numbers is (5+1)(1+1)(1+1) = 24 (Note: This doesn’t include your condition of them summing to an even number). This is because for any prime that appears [math]n[/math] times, you can make it appear anywhere between [math]0[/math] and [math]n[/math] times in the first number, for a total of [math]n+1[/math] choices (and the second number just takes whatever prime factors are unused). So in general, if you wanted to figure out how many ways of creating two numbers that multiply to [math]p_1^{a_1}p_2^{a_2}....p_n^{a_n}[/math], where [math]p_i[/math] is a prime number and [math]a_i[/math] is how often it occurs, it would be [math](a_1 + 1)(a_2 + 1)....(a_n + 1)[/math]Hope this helped.
How do i find the sum of the first 499 even Natural numbers?
In Excel you could just put 2 in A1, then 4 in A2. Now select those cells and copy them down to row 499. Put the sum formula =SUM(A1:A499) in A500 Done! Or you can use math. The first 499 even natural numbers are: 2, 4, 6, ..., 992, 996, 998 If you notice, the first and last pair up to be 1000: 2 + 998 = 1000 You can do the same with the next lowest and highest: 4 + 996 = 1000 And then the next: 6 + 994 = 1000 You can repeat this process for 249 pairs. You'll be left with 500 in the middle without a mate. So that is 249 x 1000 + 500 = 249,500 You can also think of it as 249½ pairs of 1000. 249.5 x 1000 = 249,500 Answer: The sum of the first 499 even natural numbers is 249,500
An odd number of numbers multiplied gives you a negative right?
im trying to remember these negative and positive rules when multiplying but idk where to go to find that info lol. basically what I'm asking is if you're multiplying a random amount of negative numbers, an even amount means it will be positive, and an odd number means it will be negative? And if there are both positive and negative numbers being multiplied, how would you go about firguring it out? Thanks!!!
Why does a negative number multiplied with another negative number give a positive number as a product?
Let's say you are playing a game involving black and red chips. At the end of the game, for each black chip that you have, you receive one dollar (+1). For each red chip that you have, you have to pay one dollar (-1). Now, these chips are packed together in bags of five, and say at some point in the game you've got several bags of black chips and several bags of red chips.If someone gives you three bags of black chips, then you gain 15 dollars. (3)(5)=15.If someone takes away three of your bags of black chips, then you lose 15 dollars. (-3)(5)=-15.If someone gives you three bags of red chips, then you lose 15 dollars. (3)(-5)=-15.If someone takes away three of your bags of red chips, then you gain 15 dollars. (-3)(-5)=15.The key idea is that negative numbers represent changes, not amounts. It doesn't make sense to say that you have -4 slices of bread. It does, however, make sense to say that you ate 4 slices of bread, and therefore the change in the number of slices you have is -4.
What 3 numbers can be multiplied to get 36?
The distinct integers 2, 3, and 6.
How many positive 3 digit integers have an odd number of divisors?
Hint: To have an odd number of divisors, an integer must be a perfect square. 10^2 = 100 11^2 = 121 ... 31^2 = 961 32^2 = 1024 (discard) So, the total number of the positive 3-digit integers is 31-10+1 = 22.