Question

which two consecutive even number have an lcm of 180

Answer 1

Let’s assume the two terms:1st term : 2x2nd term : 2x + 2There is a rule which states thatIf ‘a’ and ‘b’ are two numbers, then their product is equal to the product of their LCM and HCF.i.e,a∗b=HCF∗LCM - (i)We can logically say that the HCF of any two consecutive even numbers is always equal to ‘2’ provided that the smallest no. is at least ‘2’.i.e.,HCF of ‘2x’ and ‘2x + 2’ is ‘2’ - (ii)Therefore,from (i) and (ii),2x∗(2x+2)= HCF * LCM2x∗(2x+2)=2∗180x∗(2x+2)=1802x2+2x−180=0Solving the quadratic equation we get,x=9,−10(we discard -10 since it is negative)Therefore, the numbers are2∗9=18 (1st no.)2∗9+2=18+2=20 (2nd no.)Answer : 18,20Thank You,Hope this help U

Answer 2

Let 2n and 2(n+1) be the consecutive numbers. Then
2n = 2 * n
2(n + 1) = 2 * (n + 1)
LCM = 2 * n * (n + 1)
Therefore
2n(n + 1) = 180
n(n + 1) = 90
n^2 + n - 90 = 0
(n + 10)(n - 9) = 0
n = 9, n > 0
2n = 18
2(n + 1) = 20
Numbers are 18 and 20