Three numbers 1400,1960and n, have a G.C.DandL.C.M of 70 2^2×5^2×7^2×11 respectively.find the least possible value of n
The three numbers have a G.C.D of 70, which means they are all divisible by 70. So we can write them as 70a, 70b, and 70c, where a, b, and c are integers.
The L.C.M of the three numbers is 2^2 × 5^2 × 7^2 × 11. To find the L.C.M, we need to find the highest power of each prime factor that appears in any of the numbers. Since the G.C.D is 70, none of the numbers have any prime factors other than 2, 5, 7, and 11.
Let's find the highest power of each prime factor that appears in the numbers:
For 2: The highest power that appears in any of the numbers is 2^2, since 1400 = 2^3 × 5^2 × 7 and 1960 = 2^3 × 5 × 7^2. So the highest power of 2 in the L.C.M is 2^2.
For 5: The highest power that appears in any of the numbers is 5^2, since 1400 = 2^3 × 5^2 × 7 and 1960 = 2^3 × 5 × 7^2. So the highest power of 5 in the L.C.M is 5^2.
For 7: The highest power that appears in any of the numbers is 7^2, since 1400 = 2^3 × 5^2 × 7 and 1960 = 2^3 × 5 × 7^2. So the highest power of 7 in the L.C.M is 7^2.
For 11: The highest power that appears in any of the numbers is 11^1, since none of the numbers have any factors of 11. So the highest power of 11 in the L.C.M is 11^1.
Therefore, the L.C.M of the three numbers is 2^2 × 5^2 × 7^2 × 11.
To find the least possible value of n, we need to find the least common multiple (LCM) of 70a, 70b, and 70c that gives the LCM of 2^2 × 5^2 × 7^2 × 11. In other words, we need to find the smallest value of a, b, and c such that 70a, 70b, and 70c have the prime factors of 2^2 × 5^2 × 7^2 × 11.
Since the maximum power of each prime factor in the LCM is 2^2 × 5^2 × 7^2 × 11, each of a, b, and c must have the maximum power of each prime factor, which means a = b = c = 1.
Therefore, the least possible value of n is 70.