Jonah has a large collection of marbles. He notices that if he borrows 5 marbles from a friend, he can arrange the marbles in rows of 13 each. What is the remainder when he divides his original number of marbles by 13?
Now that he has the extra 5 marbles, his number must be a multiple of 13, that is, it divides evenly by 13
So obviously before he had the 5 his number must have had a remainder of 5 when divided by 13
e.g suppose he originally had 320
when divided by 13 we get 24 rown with 5 marbles left over. So he gets 5 from his friend and now has 325 , which divides by 13 to get 25 rows.
let me try this again.
Now that he has the extra 5 marbles, his number must be a multiple of 13, that is, it divides evenly by 13
So obviously before he had the 5 his number must have had a remainder of 13-5 or 8 when divided by 13
e.g suppose he originally had 320
when divided by 13 we get 24 rown with 8 marbles left over. So he gets 5 from his friend and now has 325 , which divides by 13 to get 25 rows.
325÷13=25 R 0
324÷13 =24 R 12
323÷13=24 R 11
322÷13 = 24 R 10
321÷13 = 24 R 9
320÷13 = 24 R 8
320÷13 = 24 R 7
etc
Well, Jonah seems to be really "marble"-ous with his collection! So, if he borrows 5 marbles from a friend and arranges them in rows of 13 each, that means he has 13 marbles in each row. Now, to find the remainder when he divides his original number of marbles by 13, we just need to know how many rows of 13 he has. But since we don't know the exact number of marbles, I'm afraid the remainder will remain a mystery for now! Keep on "rolling" with those marbles, Jonah!
Let's break down the information given.
1. Jonah borrows 5 marbles, so we can assume that the number of marbles he originally had is divisible by 13, plus 5.
2. We know that if Jonah borrows 5 marbles, he can arrange the remaining marbles in rows of 13 each.
Based on this information, we can conclude that the original number of marbles Jonah had is a multiple of 13. Let's find the remainder when this multiple is divided by 13.
To find the remainder, we can subtract the borrowed marbles (5) from the original number of marbles Jonah had and then divide this difference by 13.
Let "x" be the original number of marbles Jonah had.
So, x - 5 is divisible by 13.
Now, let's find the remainder when (x - 5) is divided by 13.
Using modular arithmetic, we can write this as:
(x - 5) ≡ 0 (mod 13)
Now, add 5 to both sides of the congruence:
(x - 5) + 5 ≡ 0 + 5 (mod 13)
This simplifies to:
x ≡ 5 (mod 13)
So, when Jonah divides the original number of marbles he had by 13, the remainder is 5.
To find the remainder when Jonah divides his original number of marbles by 13, we need to understand the concept of divisibility.
Let's assume Jonah initially had "x" marbles.
When Jonah borrows 5 marbles, he is left with "x - 5" marbles.
Now, we are told that if Jonah arranges the marbles in rows of 13 each, there is no remainder. This means the total number of marbles must be divisible evenly by 13.
To represent this mathematically, we can set up the equation:
(x - 5) ÷ 13 = 0
To solve for "x," we can multiply both sides of the equation by 13:
x - 5 = 0
Adding 5 to both sides of the equation, we get:
x = 5
So, Jonah initially had 5 marbles.
To find the remainder when 5 is divided by 13, we can perform the division:
5 ÷ 13 = 0 remainder 5
Therefore, the remainder when Jonah divides his original number of marbles by 13 is 5.