Answer :
(a) To find the other number when the H.C.F. and L.C.M. are given, we use the formula:
[tex]\text{H.C.F.} \times \text{L.C.M.} = \text{Product of the two numbers}[/tex]
Given H.C.F. is 6 and L.C.M. is 36, one number is 18.
[tex]6 \times 36 = 18 \times x[/tex]
So, the other number, [tex]x[/tex], is
[tex]x = \frac{6 \times 36}{18} = 12[/tex]
Thus, the other number is 12.
(b) Given the H.C.F. of [tex]x[/tex] and 210 is 30, and their L.C.M. is 2310, use the same formula:
[tex]30 \times 2310 = x \times 210[/tex]
To find [tex]x[/tex]:
[tex]x = \frac{30 \times 2310}{210} = 330[/tex]
Thus, the value of [tex]x[/tex] is 330.
(c) Given the H.C.F. of 45 and 60 is 15, we use the formula:
[tex]\text{L.C.M.} = \frac{45 \times 60}{15}[/tex]
Calculating gives:
[tex]\text{L.C.M.} = 180[/tex]
So, their L.C.M. is 180.
(d) Given the H.C.F. is 14 and product of two numbers is 1260, use:
[tex]\text{L.C.M.} = \frac{\text{Product of the numbers}}{\text{H.C.F.}}[/tex]
[tex]\text{L.C.M.} = \frac{1260}{14} = 90[/tex]
So, their L.C.M. is 90.
(e) Given the product of two numbers is 448 and their L.C.M. is 112, we find the H.C.F. using:
[tex]\text{H.C.F.} = \frac{448}{112} = 4[/tex]
Thus, their H.C.F. is 4.
