Find the smallest square number that is divisible by each of the following number 49 and 10

Question Answer
1

find the smallest square number that is divisible by each of the following number 49 and 10

2

Find the smallest square number divisible by 49 and 10

:light_bulb: RULE / FORMULA USED:
To find the smallest square number divisible by a set of numbers, we first find the Least Common Multiple (LCM) of the given numbers. Then, we adjust the prime factors of the LCM so that every prime factor has an even exponent, making the result a perfect square.

:brain: SOLUTION STEPS:

Step 1 — Prime Factorization of the Numbers
First, we break down 49 and 10 into their prime factors:

  • 49 = 7 \times 7 = 7^2
  • 10 = 2 \times 5

Step 2 — Find the LCM of 49 and 10
The LCM is found by taking the highest power of each prime factor present in the numbers:

  • LCM(49, 10) = 2^1 \times 5^1 \times 7^2
  • LCM = 2 \times 5 \times 49 = 490

Step 3 — Convert the LCM into a Perfect Square
A number is a perfect square only if the exponent of every prime factor is even. Let’s look at the prime factorization of our LCM (490):

  • 490 = 2^1 \times 5^1 \times 7^2

The prime factors 2 and 5 have odd exponents (1). To make them even, we must multiply the LCM by 2 and 5:

  • \text{Smallest Square} = (2^1 \times 2^1) \times (5^1 \times 5^1) \times 7^2
  • \text{Smallest Square} = 2^2 \times 5^2 \times 7^2

Step 4 — Final Calculation
Now, we calculate the final value:

  • 4 = 2^2
  • 25 = 5^2
  • 49 = 7^2
  • 4 \times 25 \times 49 = 100 \times 49 = 4900

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:white_check_mark: ANSWER: The smallest square number divisible by 49 and 10 is 4900.
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:bullseye: KEY CONCEPTS:

1. Least Common Multiple (LCM)

  • Definition: The smallest positive integer that is divisible by both a and b.
  • In this problem: We used it to find the baseline number (490) that satisfies the divisibility condition before addressing the “square” condition.

2. Perfect Square

  • Definition: An integer that is the square of an integer; its prime factorization must contain only even exponents.
  • In this problem: We multiplied 490 by 10 (the product of missing prime pairs) to reach the smallest square (4900).

:warning: COMMON MISTAKES:

:cross_mark: Confusing LCM with the Final Answer

  • Wrong: Thinking 490 is the answer because it is the smallest number divisible by both.
  • Right: 490 is not a square. You must ensure the exponents are even.
  • Why it’s wrong: The question specifically asks for a square number.

Would you like me to generate a similar practice problem with three numbers to test your skills?