I have some beads. When I divide all the beads into 4 equal groups, 1 bead is left. When I divide one such group into 4 equal groups again, the remainder will still be 1. Wen I split one such group into 4 equal groups again, the remainder is still 1. What is the minimum number of beads I have?

Question

I have some beads. When I divide all the beads into 4 equal groups, 1 bead is left. When I divide one such group into 4 equal groups again, the remainder will still be 1. Wen I split one such group into 4 equal groups again, the remainder is still 1. What is the minimum number of beads I have?

Answer 1

4n+1

--> 4(4n+1) + 1 = 4n + 5
---- >4(4n+5) + 1 = 16n + 21
----> 4(16n + 21) + 1 = 64n + 84 + 1 = 64n + 85

let n = 0
we get 85

testing:
85 ÷ 4 = 21 , rem 1
21 ÷ 4 = 5 , rem 1
5 ÷ 4 = 1 , rem 1

85 looks good

Answer 2

--> 4(4n+1) + 1 = 4n + 5

could you please explain this step.

Regards
Raghu

Answer 3

Your answer should substitute n = 1 and your equation was wrong in the first 2 statements ...

4a + 1 = Total beads
4b + 1 = a
4c + 1 = b
replace b with C in second equation 4(4c + 1 ) + 1 = a
16c + 5 = a
replace a with c in statement 1
4(16c + 5) + 1 = a.
64c + 21 = a.
Minmum number for c = 1.
then a = 85

Answer 4

To find the minimum number of beads you have, we can use a method called working backwards.

Let's start with the last step. When you split one of the groups into 4 equal groups again, there is still a remainder of 1. This means that the number of beads in that group is (4 x N) + 1, where N is a whole number.

If we move one step back before this, we can see that when you divide one group into 4 equal groups, there is still a remainder of 1. This means the number of beads that were in that group before dividing it into smaller groups is (4 x N) + 1.

Now, let's move back one more step. When you initially divided all the beads into 4 equal groups, 1 bead was left. This means the total number of beads before dividing them into smaller groups was (4 x N) + 1.

To find the minimum number of beads, we need to find the smallest value of N for which (4 x N) + 1 satisfies all the conditions.

If we plug in the values of N starting from 0, we can find the smallest value that gives us the desired remainder:

When N = 0, (4 x N) + 1 = 1 (remainder is not 1)
When N = 1, (4 x N) + 1 = 5 (remainder is 1)
When N = 2, (4 x N) + 1 = 9 (remainder is not 1)
When N = 3, (4 x N) + 1 = 13 (remainder is not 1)
...

By trying different values of N, we can see that the smallest value that gives us a remainder of 1 is when N = 1.

Therefore, the minimum number of beads you have is (4 x 1) + 1 = 5.