Question

The number $4,000,000$ has $63$ positive integral factors. How do you find $a$ and $b$, where $2{a^2}5{b^2}$ is the product of all positive factors of $4,000,000$?

Answer 1

Hint:First we will mention the term square of a number. Then evaluate the square of the number. For evaluating the square of the term, we will be using the property of exponent which is given by ${a^n} \times {a^m} = {a^{n + m}}$. Break the given term and substitute the terms in the formula to evaluate the square.

Complete step by step answer:
We will start off by explaining the term square of a term.
So, to square a number means to just multiply it by itself.
Now we will factorise the term $4,000,000$ in terms of $10$ and $4$.
$
= 4,000,000 \\
= 4 \times 1,000,000 \\
= 4 \times {10^6} \\
$
Now we will break the term $4 \times {10^6}$ such that we get the factors in terms of $2$ and $5$.
$
= 4 \times {10^6} \\
= {2^2} \times {10^6} \\
= {2^2} \times {(2 \times 5)^6} \\
$
Now we will distribute the power between the terms inside the bracket.
$
= {2^2} \times {(2 \times 5)^6} \\
= {2^2} \times {2^6} \times {5^6} \\
= {2^{2 + 6}} \times {5^6} \\
= {2^8} \times {5^6} \\
$
Now if we compare the term ${2^8} \times {5^6}$ with the given term $2{a^2}5{b^2}$.
Hence, the values of $a$ and $b$ are $8,6$.

Note: While applying the formula, choose the operation according to the order of the rule. Use the PEMDAS rule here, to evaluate the value of the square of the term. While converting orders do not matter for addition and multiplication. But order is important for subtraction and division. Make sure that you read the statement twice before translating it to an expression. Pay extra attention to the statements where multiplication and division is involved. While cross multiplying the terms, make sure to multiply the terms along with their signs.