Answer: Indeed, 16,641 is a perfect square.
Step-by-step explanation: 16641 is the 129th perfect square number
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1 2 9
1 1 66 41
1
22 66
44
249 22 41
22 41
258 0
Number = 16641
Square Root = 129
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Step-1:
Form pairs of digits from the number starting with the digit at the one's place. Bar each pair.
1 66 41
Step-2:
Now we need to multiply a number by itself so that the product ≤ 1
Here 1×1=1≤1, So the divisor is 1 and the quotient is 1. Now perform the division and obtain the remainder.
1
1 1 66 41
1
0
Step-3:
Next, we bring down 66 and multiply the quotient 1 by 2 to get 2, starting the new divisor.
1
1 1 66 41
1
2 66
Step-4:
The new divisor's one's place digit should be 2, as multiplying 22 by 2 gives 44.
This makes the new divisor 22 and the next quotient digit is 2. Now conduct the division and compute the remainder.
1 2
1 1 66 41
1
22 66
44
22
Step-5:
Now we bring down 41 and 12 (quotient) multiplied by 2 becomes 24, starting the new divisor's first digit.
1 2
1 1 66 41
1
22 66
44
24 22 41
Step-6:
The new divisor's one's place digit must be 9 since 249 multiplied by 9 results in 2241.
This makes the new divisor 249 and the quotient’s next digit is 9. Now conduct the division and get the remainder.
1 2 9
1 1 66 41
1
22 66
44
249 22 41
22 41
0
Answer: 129 (as proof, 129^2) = 16,641 or 129 x 129 = 16,641
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Secondary Solution shortcut:
A perfect square is a number expressible as the product of two identical integers.
The only way to accurately determine if a number is a perfect square is to identify the factors. Before we exert effort into finding factors, a quick method exists to help ascertain whether further work is warranted.
A perfect square never concludes in 2, 3, 7, or 8. If your number ends with any of these digits, you can stop here as it is not a perfect square.
Obtain the digital root of your number, which is merely the sum of all the digits. If you're confused, don't fret; we’ll clarify each point further below.
Any potential perfect square numbers must possess a digital root of 1, 4, 7, or 9.
Let’s put it to the test...
Step 1:
What is the last digit of 16,641? It is 1. Is 1 part of the digits that can never be perfect squares (2, 3, 7, or 8)?
Answer: NO, 1 isn’t in the digits incapable of being perfect squares. Let's proceed.
Step 2:
We now need to compute the digital root:
Split the number into its digits and compute their total:
1 + 6 + 6 + 4 + 1 = 18
Should the outcome have more than one digit, add the digits of the resultant again:
1 + 8 = 9
What’s the digital root of 16,641?
Answer: 9
Step 3:
So now we ascertain that the digital root of 16,641 is 9. Does 9 intersect with the confirmed list of digital roots inherently square (1, 4, 7, or 9)?
Answer: YES, 9 is listed as one of the possible digital roots for perfect squares. Thus, we conclude that 16,641 might be a perfect square!
Factoring
Alright, now we know for certain that 16,641 might be a perfect square. We need to find factors to ensure this.
Here are the factors of 16,641:
1 x 16,641, 3 x 5,547, 9 x 1,849, 43 x 387, 129 x 129
The highlighted combination confirms 16,641 as a perfect square. Do you understand why? A number can only qualify as a perfect square if the product of two identical numbers equals it.
Proof: 129 x 129 = 16,641
