Question

If my age is divided by 5, the remainder is 4. If it’s divided
by 3, the remainder is 2. If it’s divided by 7, the remainder
is 5. How old am I?

Answer 1

x/5 =4
x/3 =2
x/7=5

x/5 + x/3 + x/7 =4+2+5
x(1/5 + 1/3 + 1/7)=11
x=11(1/5 + 1/3 + 1/7)

Answer 2

what is the value of x?

Answer 3

you can do what remains

Answer 4

on no. try chinese remainder theorem

Answer 5

will work it out if you like

Answer 6

I agree with satellite; the working above in nonsense.

Answer 7

is*

Answer 8

i believe its 89

Answer 9

That is meaningless; like above, use the chinese remainder theorem.

Answer 10

i believe that it is early in the morning to use crt, but it is actually interesting to list.
remainder of 4 when divided by 5
9, 14, 19, 24, 29, ... and so see what happens when you consider the remainders when dividing by 3. the patter in
0, 2, 1, 0, 2 , 1,

Answer 11

in any case method is described clearly here
http://marauder.millersville.edu/~bikenaga/numbertheory/chinese-remainder/chinese-remainder.html

Answer 12

for a really nice description of the history of the chinese remainder theorem see
"the mathematical experience"

Answer 13

so is the answer 89 ?

Answer 14

\[x = 89 + 105k \]

But realistically, k =0 is the only solution.

Answer 15

@INewton : what is k ?

Answer 16

k is an integer where
\[k \geq 0 \]

Answer 17

\[x \equiv \begin{cases} 2\mod\ 3 \\ 4\mod\ 5 \\ 5\mod\ 7\end{cases}\]
\[\text{Let } x_1 \equiv \begin{cases} 1\mod\ 3 \\ 0\mod\ 5 \\ 0\mod\ 7\end{cases} \implies x_1 \equiv \begin{cases} 1\mod\ 3 \\ 0\mod\ 35 \end{cases} \implies x_1=35x_1' \text{ for some integer } x_1'\]

\[\text{Therefore } 35x_1' \equiv 1 \mod 3 \implies 2x_1' \equiv 1\mod 3 \]
\[\text{Let's take } x_1' = 2 \implies x_1 = 70\]

Following the obvious notation we find similarly:

\[x_2 = 21 \text{ and } x_3 = 15 \]

\[\text{Clearly, } 2x_1 + 4x_1 + 5x_3 = x_{min} + 105k \implies x_{min} = 299 - 105k\]

For the smallest positive integer x we have x = 89 (k=2), be we also have solutions of the form
\[x = 89 + 105k\ \forall k \in \mathbb{N} \]

Answer 18

Ignore the typos ¬_¬ (e.g. it should be 2x_1 + 4x_2 + 5x_3)