Question

I just found a number with an interesting property:

When I divide it by 2, the remainder is 1.
When I divide it by 3, the remainder is 2.
When I divide it by 4, the remainder is 3.
When I divide it by 5, the remainder is 4.
When I divide it by 6, the remainder is 5.
When I divide it by 7, the remainder is 6.
When I divide it by 8, the remainder is 7.
When I divide it by 9, the remainder is 8.
When I divide it by 10, the remainder is 9.

It's not a small number, but it's not really big, either.

Find the smallest number with such property.

PLEASE ANSWER I WILL MARK AS BRAINLIEST. DONT SPAM​

Answer 1

Answer:

Step-by-step explanation:

we're looking for n where n divided by x (x being each number 2-10) leaves a remainder of x-1, n+1 divided by each number 2-10 leaves no remainder. To find n+1, we must simply multiply all prime factors of 2-10, excluding any duplicates. This is equal to 2*2*2*3*3*5*7 = 2520, then just subtract one to get the answer of 2519.

Answer 2

Before reading the answer can I interest you in a clue?

The key to this is to realise that if a number N (eg 104) is to leave a remainder, say 4 when divided by 5 then N + 1 (ie 105) is exactly divisible by 5, similarly if a number (eg 87) is to leave a remainder 7 when divided by 8 then N + 1 (ie 88) is exactly divisible by 8:-

Therefore in our question N + 1 is divisible by 2, 3, 4, 5, 6, 7, 8, 9 & 10

There is a tool here you can use to try different values of N

Note: no information is sent to me, the calculation is done entirely locally, on your computer.

Meaning N+1 is 2000

N+1 has the following factors:

2✓  3✗  4✓  5✓  6✗  7✗  8✓  9✗  10✓  All Factors✗  

The lowest value for N + 1 is 2520, which means the lowest value for N is:-

2519

How do we arrive at N + 1 = 2520?

As stated N + 1 is divisible by 2, 3, 4, 5, 6, 7, 8, 9 & 10 so clearly one solution would be 10! (ie 10*9*8*7*6*5*4*3*2 = 3628800 for N +1) but this is not the lowest possible. Follow the logic below...

N + 1 must be a multiple of 2

N + 1 must be a multiple of 4 but if it is a multiple of 2 & 4 it is necessarily a multiple of 8

N + 1 must be a multiple of 5 but if it is a multiple of 2 & 5 it is necessarily a multiple of 10

N + 1 must be a multiple of 6 but if it is a multiple of 2 & 9 (18) it is necessarily a multiple of 6

N + 1 must be a multiple of 7

N + 1 must be a multiple of 9

Hence 2 x 4 x 5 x 7 x 9 = 2520 is a multiple of 2, 3, 4, 5, 6, 7, 8, 9 & 10

Mathematically the process we are dealing with here is known as lowest or least common multiple.