Asif notices that $24^2 = 576$ and $2 + 4 = 6$ gives the last digit of 576. He checks two more examples: $27^2 = 729$ $2 + 7 = 9$ Last digit 9 $29^2 = 841$ $2 + 9 = 11$ Last digit 1 Asif concludes that he can find the last digit of any square number greater than 100 by adding the digits of the number being squared. Give a counter example to show that Asif's conclusion is not correct. --- Claire tells Asif that he should look only at the last digit of the number being squared. $27^2 = 729$ $7^2 = 49$ Last digit 9 $24^2 = 576$ $4^2 = 16$ Last digit 6 Using Claire's method determine the last digit of 234567892. --- Given Claire's method is correct, use proof by exhaustion to show that no square number has a last digit of 8.

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Asif notices that $24^2 = 576$ and $2 + 4 = 6$ gives the last digit of 576 - AQA - A-Level Maths Pure - Question 6 - 2022 - Paper 2

Question 6

Asif notices that $24^2 = 576$ and $2 + 4 = 6$ gives the last digit of 576. He checks two more examples: $27^2 = 729$ $2 + 7 = 9$ Last digit 9 $29^2 = 841$ $2 ... show full transcript

Worked Solution & Example Answer:Asif notices that $24^2 = 576$ and $2 + 4 = 6$ gives the last digit of 576 - AQA - A-Level Maths Pure - Question 6 - 2022 - Paper 2

Step 1

Give a counter example to show that Asif's conclusion is not correct.

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Answer

To provide a counterexample, we can use the number 13.

Calculating the square: $13^2 = 169$

Now, adding the digits of 13: $1 + 3 = 4$

However, if we consider the last digit of 169, we see: Last digit is 9.

This shows that Asif's method does not yield the correct last digit, thus proving his conclusion incorrect.

Step 2

Using Claire's method determine the last digit of 234567892.

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Answer

We only need to consider the last digit of 234567892, which is 2.

According to Claire's method: $2^2 = 4$

Thus, the last digit of 234567892 is 4.

Step 3

Use proof by exhaustion to show that no square number has a last digit of 8.

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Answer

To prove that no square number can end with a last digit of 8, we analyze the possible last digits of square numbers.

The possible last digits for any integer squared are: 0, 1, 4, 5, 6, 9.

For a number ending in 0:
- $0^2 = 0$
For a number ending in 1:
- $1^2 = 1$
For a number ending in 2:
- $2^2 = 4$
For a number ending in 3:
- $3^2 = 9$
For a number ending in 4:
- $4^2 = 16$ last digit 6
For a number ending in 5:
- $5^2 = 25$ last digit 5
For a number ending in 6:
- $6^2 = 36$ last digit 6
For a number ending in 7:
- $7^2 = 49$ last digit 9
For a number ending in 8:
- $8^2 = 64$ last digit 4
For a number ending in 9:

$9^2 = 81$ last digit 1

Since none of these calculations yield a last digit of 8, we conclude through proof by exhaustion that no square number has a last digit of 8.

Asif notices that $24^2 = 576$ and $2 + 4 = 6$ gives the last digit of 576 - AQA - A-Level Maths Pure - Question 6 - 2022 - Paper 2

Worked Solution & Example Answer:Asif notices that $24^2 = 576$ and $2 + 4 = 6$ gives the last digit of 576 - AQA - A-Level Maths Pure - Question 6 - 2022 - Paper 2

Give a counter example to show that Asif's conclusion is not correct.

Using Claire's method determine the last digit of 234567892.

Use proof by exhaustion to show that no square number has a last digit of 8.

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