Number Theory

Number Theory Level 3

What is the smallest positive integer that is evenly divisible by the first ten positive integers?

The answer is 2520.

2 solutions

Eli Ross Staff
Nov 17, 2015

Great problem, @Edu Alburo ! This can clear up some misconceptions surrounding Lowest Common Multiples .

In particular, we're looking for $\text{lcm}(1,2,3,4,5,6,7,8,9,10).$ If we write out the Prime Factorizations , this is $\text{lcm}(1,2,3,2^2,5,2\cdot 3, 7, 2^3, 3^2, 2\cdot 5) = 2^3 \cdot 3^2 \cdot 5 \cdot 7 = \boxed{2520}.$

If you're wondering how that last step worked, let's think about the prime factorization of this number which is divisible by 1 through 10. In particular, it must contain the highest powers of any primes found in the factorizations of $1,2,\ldots,10.$

Thanks for appreciating my problem dude!

Edu Alburo - 5 years, 6 months ago

our question is "What is the smallest positive integer that is evenly divisible by the first ten positive integers?" How u drew the answer 2520 that is evenly divisible by all the positive integers but not evenly divisible by 8(i.e 2520/8=315). I think answer should be 5040. That is evenly divisible by first ten positive integers. plz correct me if I am wrong.

Niaz Ghumro - 5 years, 6 months ago

What's wrong with 2520/8? Are you thinking that "evenly divisible by" means the quotient is an even number and 315 is odd? If so, "evenly divisible by" just means that the answer is an integer, not an even number.

8 just happens to be the only one that does not give an even answer because it has the highest degree of twos (2^3) and any other factor is going to leave at least one two when divided out, just like all the quotients are divisible by three except 2520/9 because that's the only one that divides out all the threes.

Eric Osborn - 5 years, 6 months ago

In Response to Eric Osborn: Then what is the difference between these two "What is the smallest positive integer that is evenly divisible by the first ten positive integers?" and "What is the smallest positive integer that is divisible by the first ten positive integers?" yes I got it as even quotient.

Niaz Ghumro - 5 years, 6 months ago

@Niaz Ghumro – Nothing, I believe it's a tautology used because some people don't get that in mathematics (unlike in common language) the term divisible already implies there is no remainder.

Henk Elemans - 5 years, 6 months ago

Plusle Atreides
Nov 20, 2015

To do it with anti-mainstream ways (LoL)

(if you don't know about LCM) first, the positive integer that divisible by 1,2,3,4,5,6,7,8,9,10 is 1x2x3x4x5x6x7x8x9x10 but it's not a smallest, so do these steps

1 \times x = x, for any number, so remove 1. ----> 2x3x4x5x6x7x8x9x10

10 = 2x5, remove 10, and mark the 2 and 5 ----> (2)x3x4x(5)x6x7x8x9

9 = 3x3, in that list already have one 3, add one 3, and mark it ----> (2)x(3)x(3)x4x(5)x6x7x8

8 = 2x4, remove 8 ----> (2)x(3)x(3)x(4)x(5)x6x7

7 prime number ----> (2)x(3)x(3)x(4)x(5)x6x(7)

6= 2x3, remove 6 ----> (2)x(3)x(3)x(4)x(5)x(7)

compute (2)x(3)x(3)x(4)x(5)x(7) = 2520.

Number Theory

The answer is 2520.

2 solutions

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