Find x, gcf of (x,40)is 1400

Answer 1

We can start by finding the prime factorization of 40:

40 = 2^3 * 5

To find the GCF of x and 40, we need to determine the common factors they share. Since 1400 is the GCF, we know that it must be a multiple of 1400.

1400 = 2^3 * 5^2 * 7

So, any value of x that shares these factors with 40 will have a GCF of 1400. We can write x as:

x = 2^a * 5^b * 7^c (where a,b,c are non-negative integers)

To guarantee that the GCF of x and 40 is 1400, we need:

a ≤ 3 (so that x has at most as many factors of 2 as 40)
b ≤ 2 (so that x has at most as many factors of 5 as 40)
c ≤ 1 (so that x has at most as many factors of 7 as 40)

So, we just need to choose values for a,b,c that satisfy these conditions:

a = 3, b = 2, c = 1

Then:

x = 2^3 * 5^2 * 7^1 = 1400

Therefore, the value of x that has a GCF of 1400 with 40 is x = 1400.