David did not notice the multiplication sign between two three-digit numbers and wrote one six-digit number, which is equal to seven times their product. find these numbers. also: let's call three prime numbers "cool" if their product is five times greater than their sum. how many "cool" triples are there?

1) We can get that number hit and trial method.

It is said that "two three-digit numbers and wrote one six-digit number, which is equal to seven times their product".

Let us check number 143.

If we write 143 two times without any sign in between , we get 143143 ( a six digit number).

But if we multiply 143 × 143 , we get 20449.

7 times of 20449 equals 143143.

2) Let us assume required prime numbers are a, b and c.

Product of a, b and c= abc.

Sum of a, b and c = a+b+c.

"Their product is five times greater than their sum."

Therefore,

abc = 5(a+b+c)  equation (1)

Now, let us take first prime numbers 2 and second 5.

Plugging a=2 and b=5.

2×5×c = 5(2+5+c).

10c = 5(7+c)

10c = 35 +5c.

Subtracting 5c from both sides, we get

10c-5c = 35 +5c-5c.

5c = 35.

Dividing both sides by 5, we get

c=7.

Therefore, first "cool" triple is 2,5,7.

Let us check by taking a=2 and b=7.

Plugging a=2 and b=7 in equation (1), we get

2×7×c = 9(2+7+c).

c=9. But it's not a prime number.

Let us take a=2 and b=11, we get

2×11×c = 11(2+11+c).

c=13   (A prime)

If we take a=2 and b=17, we get

2×17×c = 17(2+17+c).

c=19   (A prime).

(-1,-0.5)

step-by-step explanation:

-1/2 +2

step-by-step explanation: