A Number N Divides Each Of 17 And 30 With The Same Remainder In Each Case. What Is The Largest Value N Can Have?
Since we're finding the largest value of n, this means n is 13 and furthermore the largest value that n can have when it divides both 17 and 30 and obtains the same remainder is 13.
Oops. N=13
To find the largest value N that divides both 17 and 30 with the same remainder, we need to identify the common remainder.
Let's see how we can approach this problem step by step:
Step 1: Find the remainders when dividing 17 and 30 by N.
To find the remainders, we divide each number by N:
- 17 ÷ N = Q₁ with a remainder R₁
- 30 ÷ N = Q₂ with a remainder R₂
Step 2: Determine the relationship between the remainders.
Since N divides both 17 and 30 with the same remainder, we can set up an equation using the remainders:
R₁ = R₂
Step 3: Consider the possible remainders.
Both remainders R₁ and R₂ must be positive values and less than N. So we need to find the factors of both 17 and 30.
Factors of 17: 1, 17
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
The common remainders should be one of these factors.
Step 4: Test the possible remainders.
We start testing each possible remainder by dividing both 17 and 30 by the factors in common:
When R = 1:
- 17 ÷ 1 = 17 with a remainder of 0
- 30 ÷ 1 = 30 with a remainder of 0
When R = 17:
- 17 ÷ 17 = 1 with a remainder of 0
- 30 ÷ 17 ≈ 1 with a remainder of 13
We can see that R = 1 is a common remainder for N, while R = 17 does not satisfy the condition.
Step 5: Determine the largest value N.
To find the largest value for N, we look for the largest factor of 17 that divides 30.
The largest factor of 17 that divides 30 is 1.
Therefore, the largest value N can have is 1.