Need Desperate help with VERY tricky math problem..!?

Question

What is the least positive integer meeting each of the following conditions?
Dividing by 7 gives a remainder of 4
Dividing by 8 gives a remainder of 5
Dividing by 9 gives a remainder of 6.

If possible, don't tell me the answer rather tell me how to figure it out!

I did the multiples and add 4,5 or 6 one except I got up to like 7 times 300 plus 4 and couldn't find one!

umm gr.8 excel math...........

Answer 1

here's another way to think about it:
We are looking for n where:
1) n=7a+4
2) n=8b+5
3) n=9c+6

1) and 2) are going to be satisfied every 8th a and every 7th b (not necessarily counting from zero)

2) and 3) are going to be satisfied every 9th b and every 8th c

So all of them will be satisfied every 63rd b (7x9). But there is an obvious negative solution, where a=b=c= -1, so that's the place to start counting.

Then b=62 and n = 501

Answer 2

I was thinking about your problem last night and came up with an elegant solution on my way to work this morning.

to satisfy the first condition:
(N+3)/7 is an integer

to satisfy the second condition
(N+3)/8 is an integer

to satisfy the third condition
(N+3)/9 is an integer

The smallest value of (N+3) that satisfies all of these is 7 x 8 x 9
therefore N=501 (the other poster is correct)

proof:

let M = N+3

as M is a positive composite number

M = f1 x f2 x f3 ...

where fn are positive prime numbers

M is divisible by 7 so at least one fn is 7
M is divisible by 8 so at least three fn are 2
M is divisible by 9 so at least two fn are 3

therefore M >= 7 x 2 x 2 x 2 x 3 x 3

Answer 3

Let n be our number.
Then n = 4 + 7t for some t.
We want 4 + 7t to be of the form 8u + 5.
So what does t have to look like?
If we plug in t = a + 8u for a between 0 and 8
and try all a we find that t = 7+ 8u.
so n = 4 + 7(7+8u) = 53 + 56u.
So 53 is the smallest number satisfying the first 2 conditions
Another way to get it.
48 = 6*8+5
49 = 7*7+4
so again we get 53.
Now we want 53+56u to be of the form 6v+9,
so what must u look like?
Again we plug in a + 9v and try them all
and find that a = 8.
So the smallest n that works is 53 + 56*8 = 501.

Answer 4

Ok this is definately no solution.

There is NO WAY that you can divide a number by 7 and come out with a smaller remainder then when you divide it by 8.

Like if you divide 11 by 7 the remainder is 4 and if you divide the 11 by 8 the remainder is 3. So, no solution.

Answer 5

List the multiples of 7 and add 4 to each.
List the multiples of 8 and add 5.
List the multiples of 9 and add 6.

Find the lowest number that is on all 3 lists.