When $n$ is divided by 10, the remainder is $a$. when $n$ is divided by 13, the remainder is $b$. what is $n$ modulo 130, in terms of $a$ and $b$?

If

N = a (mod 10)

N = b (mod 13)

gcd(10,13) = 1

then

N = 10 bx + 13 ay (mod 130)

Where

10x + 13y = 1

-> (10x + 13) (mod 2) = 1 (mod 2)

-> y (mod 2) = 1

y = -3, x = 4

-> N = 40b – 39a (mod 130)

It is given that  ra + sb should be non-negative:

N = 40b – 39a (mod 130)

N = 40b + (130 – 39)a (mod 130)

N = 40b + 91a (mod 130)

Therefore, N modulo 130, in terms of a and b is: N = 40b + 91a (mod 130).

rick has $1400 and sally has $1500

step-by-step explanation:

to find amount of money rick has after 10 years :

principal = $1000 , time = 10 years , rate = 4% annually

to find amount of money sally has after 10 years :

her friend gives to pay her $50 per year, therefore money she has after 10 years = 50 × 10 = $500

also, sally gets $1000 after 10 years,

therefore, total amount of money sally has after 10 years = 500 + 1000 = $1500

so, after 10 years : rick has $1400 and sally has $1500