a. 33
b. 34
c. 35
d. 36
e. 37
b. 34
c. 35
d. 36
e. 37
so 16+17+18+19=70 (ie, 17.5 x 4).
so I count 31+5 integers.
The sum of integers in set M can be calculated using the formula for the sum of an arithmetic series:
Sum = (N/2) * (first term + last term),
where N is the number of terms, and the first term and the last term denote the first and last numbers in the series, respectively.
In this case, the first term is -15, and the last term is y.
We are given that the sum of all the integers in set M is 70. So we can set up the equation:
70 = (N/2) * (-15 + y).
Now let's solve for y. Multiply both sides by 2:
140 = N * (-15 + y).
Next, distribute:
140 = -15N + Ny.
Rearrange the equation:
Ny - 15N = 140.
Factor out N:
N(y - 15) = 140.
Now, we need to factorize 140 to find its divisors. The factors of 140 are: 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140.
We can substitute these factors into the equation for N(y - 15) = 140 to find values for N and y. Let's check each factor:
For N = 1, 1(y - 15) = 140. The only solution for y in this case is y = 155.
For N = 2, 2(y - 15) = 140. The only solution for y in this case is y = 85.
For N = 4, 4(y - 15) = 140. The only solution for y in this case is y = 55.
For N = 5, 5(y - 15) = 140. The only solution for y in this case is y = 47.
For N = 7, 7(y - 15) = 140. The only solution for y in this case is y = 35.
For N = 10, 10(y - 15) = 140. The only solution for y in this case is y = 29.
For N = 14, 14(y - 15) = 140. The only solution for y in this case is y = 25.
For N = 20, 20(y - 15) = 140. The only solution for y in this case is y = 20.
For N = 28, 28(y - 15) = 140. The only solution for y in this case is y = 17.
For N = 35, 35(y - 15) = 140. The only solution for y in this case is y = 16.
For N = 70, 70(y - 15) = 140. The only solution for y in this case is y = 17.
From these solutions, we can see that the pair (N, y) is (70, 17).
Therefore, there are 70 numbers in the set M.
Let's calculate the number of integers between -15 and y, inclusive:
There are 15 integers from -15 to -1 (including both endpoints).
There is an additional integer for y.
So, the number of integers in set M is 15 + 1 = 16.
Next, let's set up the equation to solve for y:
-15 + (-14) + (-13) + ... + y = 70
To simplify the expression, we can group the consecutive integers from -15 to y:
(-15 + (-14) + (-13) + ...) + y = 70
To find the sum of the consecutive integers, we can use the formula for the sum of an arithmetic series:
Sum = (n / 2) * (first term + last term)
In this case, the first term is -15, and the last term is y.
Substituting these values into the equation, we get:
[(n / 2) * (-15 + y)] + y = 70
Now, let's solve for y:
[(16 / 2) * (-15 + y)] + y = 70
8 * (-15 + y) + y = 70
-120 + 8y + y = 70
9y - 120 = 70
9y = 190
y = 190 / 9
y ≈ 21.11
Since y represents an integer in this problem, we can round it up or down to the nearest whole number. Therefore, y = 21.
Thus, the number of integers in set M is 16.