When 14 is divided by 5, the remainder is 4. When 14 is divided by a positive integer n, the remainder is 2. For how many different values of n is this possible.
Is listing it out the only way?
14/2 =7
14/3 = 4 2/3 yes
14/4 = 3 2/4 yes
14/5 = 2 4/5
14/6 = 2 2/6 yes
14/7 = 2
14/8 = 1 6/8
14/9 = 1 5/9
14/10 = 1 4/10
14/11 = 1 3/10
14/12 = 1 2/12 yes
the end
I got 3
No, listing out all the possibilities is not the only way to solve this problem. Another way to approach this problem is by using modular arithmetic and number theory concepts.
Let's analyze the problem step by step:
Step 1: When 14 is divided by 5, the remainder is 4.
This can be represented as:
14 ≡ 4 (mod 5)
Step 2: When 14 is divided by a positive integer n, the remainder is 2.
This can be represented as:
14 ≡ 2 (mod n)
To find the different values of n that satisfy both equations, we can manipulate the congruences using modular arithmetic.
Step 3: Subtracting the congruence from step 1 from the congruence from step 2, we get:
14 - 14 ≡ 2 - 4 (mod n)
0 ≡ -2 (mod n)
Step 4: Simplifying the congruence from step 3 by adding n to both sides:
2 ≡ 0 (mod n)
Step 5: Since the left side (2) is less than or equal to n and greater than or equal to 0, it means that n must be greater than or equal to 2.
Therefore, the possible values of n can be any positive integer greater than or equal to 2.
To summarize, for any positive integer n greater than or equal to 2, the remainders when dividing 14 by n will be 2. Hence, there are infinitely many values of n that satisfy the given conditions.
No, listing it out is not the only way to solve this problem. There is a more systematic approach using modular arithmetic.
Let's break down the problem:
1. When 14 is divided by 5, the remainder is 4. This can be written as: 14 ≡ 4 (mod 5).
2. When 14 is divided by a positive integer n, the remainder is 2. This can be written as: 14 ≡ 2 (mod n).
To find the possible values of n, we need to consider all positive integers n for which the conditions hold true simultaneously.
Using modular arithmetic, we can rewrite the equations as:
14 ≡ 4 (mod 5) --> 14 - 4 ≡ 0 (mod 5) --> 10 ≡ 0 (mod 5),
and
14 ≡ 2 (mod n).
Now, let's focus on the second equation: 14 ≡ 2 (mod n). Here, we are looking for values of n that satisfy this equation.
To solve this, we can subtract 2 from both sides of the equation and rewrite it as:
12 ≡ 0 (mod n).
The equation 12 ≡ 0 (mod n) means that n divides 12 without leaving any remainder.
We can determine the possible values of n by finding the divisors of 12: 1, 2, 3, 4, 6, and 12.
Therefore, based on the divisors of 12, there are six different positive integer values for n that can satisfy the given conditions.
So, the answer to the question is that it is possible for six different values of n to satisfy the given conditions.