When Juan divides each of five consecutive integers by his age , the sum of five remainders he gets is 32 . When Nikita ,several years older , divides each of a different set of five consecutive integers by her age , the sum of the five remainders she gets is also 32 . What is the sum of the ages of Juan and Nikita ?
Let's break down the problem step by step.
First, we need to define some variables. Let's say Juan's age is J, and Nikita's age is N. We know that Nikita is several years older than Juan, so we can write N = J + X, where X represents the age difference between them.
Next, we need to find the consecutive integers that Juan and Nikita are dividing. Let's assume the first integer is K.
For Juan's case, we have the five consecutive integers: K, K+1, K+2, K+3, K+4. When Juan divides each of these integers by his age J, he gets remainders R1, R2, R3, R4, and R5.
Thus, we can write the equation: (K % J) + ((K+1) % J) + ((K+2) % J) + ((K+3) % J) + ((K+4) % J) = 32.
Next, let's consider Nikita's case. The five consecutive integers she divides are: K, K+1, K+2, K+3, K+4. However, since Nikita is X years older than Juan, her remainders will be different. Let's call them S1, S2, S3, S4, and S5.
The equation for Nikita's case will then be: (K % N) + ((K+1) % N) + ((K+2) % N) + ((K+3) % N) + ((K+4) % N) = 32.
Now, the problem asks us to find the sum of the ages of Juan and Nikita, so we need to calculate J + N.
To further solve the problem, we would need to find the values of J, N, and X by solving the system of equations formed by the two equations described above. However, without additional information or specific values for the remainders and consecutive integers, we cannot determine the exact sum of their ages.