Question

If the lcm of two numbers a and b is 40 what is the lcm of 7 a^2 and 7b^2

Answer 1

The LCM of 7a^2 and 7b^2, given the LCM of a and b is 40, is calculated by incorporating the square of 7 (since both numbers are multiplied by 7 and then squared) and maintaining the highest power of any prime factors from the original LCM. Therefore, the LCM of 7a^2 and 7b^2 is 7^2 × 2^3 × 5 = 1960.

Step-by-step explanation:

To find the least common multiple (LCM) of 7a2 and 7b2, given that the LCM of a and b is 40, we first recognize that the LCM of two numbers captures all prime factors involved to their highest powers. For two numbers, a and b, with an LCM of 40, we must decompose 40 into its prime factors which are 23 × 5. Since 40 = 23 × 5 is the LCM, both a and b divide 40 but neither contains prime factors not in 40.

Now, multiplying a number by 7 and squaring does not introduce any new prime factors besides 7, so we need to include 72 in our calculation for the new LCM. The new numbers are squared as well, so we must also consider the square of any prime factors already present in the original LCM.

Hence, the LCM of 7a2 and 7b2 is given by multiplying each prime factor of the original LCM to its highest power observed in the new set of numbers, plus the prime factor of 7 squared, which results in 72 × 23 × 5 = 49 × 8 × 5 = 1960. This is the LCM of 7a2 and 7b2.

Answer 2

11,200

Step-by-step explanation:

Given that the lcm of a and b is 40 then a and b are factors of 40. All the multiples of 40 as sets are 1 and 40, 2 and 20, 4 and 10, 5 and 8. Of all these, only 5 and 8 would give a lcm of 40.

Let a = 5 and b = 8 as such,

7 a^2 = 7 * 5^2 = 175

7b^2 = 7 * 8^2 = 448

Then the lcm of 7 a^2 and 7b^2

= 11,200