Math Problem Statement
How many values of satisfy both of the following conditions? (a) is a multiple of not necessarily positive. (b) is less than .
Solution
To solve the problem of finding how many values of satisfy the given conditions:
Conditions:
- (a) is a multiple of 7 (not necessarily positive).
- (b) is less than 100.
Step 1: Represent as a multiple of 7
Since is a multiple of 7, we can write , where is an integer (positive, negative, or zero).
Step 2: Find the upper limit for
We are given that must be less than 100, i.e., . Substituting into this inequality:
Dividing both sides by 7:
Thus, the largest integer value for is .
Step 3: Find the lower limit for
Since can be any multiple of 7 (not necessarily positive), can also be negative. We have no lower bound on the value of , but since is an integer, we can consider how low can go. We simply need to find when becomes negative while still less than 100.
So must satisfy the inequality . If , . As decreases, continues to decrease indefinitely.
Step 4: Count the valid values of
Since can be negative, we need to find the largest and smallest values of that satisfy the condition . From Step 2, we know the largest integer is 14.
Since can be any negative integer, the set of valid values of extends indefinitely. Therefore, ( \boxed{-7} gives us an final boundaries therefore adj
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Math Problem Analysis
Mathematical Concepts
Number Theory
Inequalities
Formulas
x = 7k, where k is an integer
7k < 100
Theorems
-
Suitable Grade Level
Grades 8-10