Determine Least Integer for Perfect Square
To find the least integer that, when multiplied by 2940, results in a perfect square, follow these steps:
Step 1: Prime Factorize 2940
Find the prime factors of 2940:
$2940 = 2 \times 1470 = 2 \times 2 \times 735 = 2^2 \times 3 \times 245 = 2^2 \times 3 \times 5 \times 49 = 2^2 \times 3^1 \times 5^1 \times 7^2$
Step 2: Analyze Prime Factor Exponents
A perfect square requires all prime factors to have even exponents.
In the factorization $2^2 \times 3^1 \times 5^1 \times 7^2$, the factors 3 and 5 have odd exponents (1).
Step 3: Calculate Smallest Multiplier
Multiply by the prime factors with odd exponents to make their exponents even.
Least multiplier = $3^1 \times 5^1 = 3 \times 5 = 15$.
Result
The least integer required is 15.
Important Questions from Integers
- A two digit number is five times the sum of its digits. If 9 is added to the number, the digits interchange their positions. The sum of digits of the number is :
If 6 times the larger number is divided by the smaller number, we get 7 as quotient and 7 as remainder. Also, if nine times the smaller number is divided by the larger number, we get 7 as the quotient and 21 as the remainder. What are the two numbers?
- Which one of the following is the greatest number by which the product of three consecutive even numbers would be exactly divisible ?
- What is the natural number n for which $3^9 + 3^{12} + 3^{15} + 3^n$ is a perfect cube of an integer ?
If the number
$2^2 \times 5^4 \times 4^6 \times 10^8 \times 6^{10} \times 15^{12}$
$\times 8^{14} \times 20^{16} \times 10^{18} \times 25^{20}$
is divisible by $10^n$, then which one of the following is the maximum value of $n$?
