-The product of all the positive factors of 20 ( including 20) can be written2^x.5^y. Find x+y.
-The number 18 is equal to 2 times the sum of its digits: 1+8=9. 2x9=18. There are 4 two digit numbers that are equal to 4 times the sum of their digitss . Find 3 of those 4.
To find the product of all the positive factors of 20, we need to list out the factors and compute their product.
The factors of 20 are: 1, 2, 4, 5, 10, and 20.
Now we can find the product:
Product = 1 * 2 * 4 * 5 * 10 * 20 = 800
Now, let's express the product 800 in the form of 2^x * 5^y.
Breaking down 800 into its prime factors, we have:
800 = 2 * 2 * 2 * 2 * 5 * 5 = 2^4 * 5^2
Therefore, x = 4 and y = 2.
Now to find x + y:
x + y = 4 + 2 = 6
So, x + y = 6.
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Now, let's move on to the second question.
To find two-digit numbers that are equal to four times the sum of their digits, we can express it as an algebraic equation.
Let the tens digit be represented by 'T' and the units digit be represented by 'U'.
The number can then be written as 10T + U.
According to the given condition, 10T + U = 4(T + U).
Simplifying the equation, we get:
10T + U = 4T + 4U
Rearranging the terms:
6T = 3U
This implies that the tens digit is half of the units digit.
Let's list down the two-digit numbers fitting this condition:
1. T = 1, U = 2 (12)
2. T = 2, U = 4 (24)
3. T = 3, U = 6 (36)
4. T = 4, U = 8 (48)
Therefore, three of the four two-digit numbers satisfying this condition are 12, 24, and 36.