SolveTheProblem

The prime numbers greater than 10 and less than 20 are 11, 13, 17, and 19. The product of any two of these primes is less than 400.

Let’s list the possible products:

11 * 13 = 143

11 * 17 = 187

11 * 19 = 209

13 * 17 = 221

13 * 19 = 247

17 * 19 = 323

So, there are 6 different products that Syd could have ended up with. Please note that 210 is not a product of two prime numbers greater than 10, which is why it was incorrect. I hope this helps! 😊

Let’s solve this step by step:

First, let’s consider the case where

t

is an integer. In this case,

⌊t⌋=t

and

⌊2t⌋=2t

. Substituting these into the equation gives

t=3t+4−2t

, which simplifies to

t=−4

. However,

−4

is not a solution because when we substitute

−4

into the original equation, we get

−4=−8+4−(−8)

.

Now, let’s consider the case where

t

is not an integer. In this case,

t

can be written as

t=n+r

where

n

is an integer and

0≤r<1

. Substituting this into the equation gives

n=3n+3r+4−2n−2r

, which simplifies to

n=−4

and

r=0

. However,

r=0

implies that

t

is an integer, which contradicts our assumption that

t

is not an integer.

Therefore, there are no values of

t

that satisfy the given equation. The equation

⌊t⌋=3t+4−⌊2t⌋

has no solutions.

The function you’ve given is

f(x)=⌊3x+12−3x​⌋

where the notation

⌊y⌋

represents the greatest integer less than or equal to

y

.

Let’s evaluate this function for a few values:

For

x=1

,

f(1)=⌊3(1)+12−3(1)​⌋=⌊4−1​⌋=−1

because the greatest integer less than

−1/4

is

−1

.For

x=2

,

f(2)=⌊3(2)+12−3(2)​⌋=⌊7−4​⌋=−1

because the greatest integer less than

−4/7

is

−1

.

You can see a pattern here. The function

f(x)

is always equal to

−1

for all positive integers

x

. This is because the numerator

2−3x

is always negative and the denominator

3x+1

is always positive for all positive integers

x

. The fraction is thus a negative number between

0

and

−1

, and the floor of this number is always

−1

.

Therefore, the sum

f(1)+f(2)+f(3)+⋯+f(999)+f(1000)

is simply

−1

added to itself

1000

times, which equals

−1000

. So, the value of the given expression is

−1000

.