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If 7272 can be written as 𝑎𝑏×𝑏𝑎a b ×b a , then find the values of 𝑎a and 𝑏b.𝑎=1,𝑏=3a=1,b=3𝑎=2,𝑏=3a=2,b=3𝑎=1,𝑏=2a=1,b=2𝑎=5,𝑏=2a=5,b=2

Question

If 7272 can be written as 𝑎𝑏 × 𝑏𝑎, then find the values of 𝑎 and 𝑏.

  • 𝑎 = 1, 𝑏 = 3
  • 𝑎 = 2, 𝑏 = 3
  • 𝑎 = 1, 𝑏 = 2
  • 𝑎 = 5, 𝑏 = 2
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Solution

1. Break Down the Problem

We need to express the number 7272 in the form of ab×ba ab \times ba , where a a and b b are digits. The digits form two two-digit numbers: ab ab (which represents 10a+b 10a + b ) and ba ba (which represents 10b+a 10b + a ).

2. Relevant Concepts

The expression can be expanded as: ab×ba=(10a+b)(10b+a) ab \times ba = (10a + b)(10b + a) We need to determine a a and b b such that: (10a+b)(10b+a)=7272 (10a + b)(10b + a) = 7272

3. Analysis and Detail

Let's denote:

  • ab=10a+b ab = 10a + b
  • ba=10b+a ba = 10b + a

Multiply these two expressions: (10a+b)(10b+a)=100ab+10a2+10b2+ab=100ab+10a2+10b2+ab=101ab+10(a2+b2) (10a + b)(10b + a) = 100ab + 10a^2 + 10b^2 + ab = 100ab + 10a^2 + 10b^2 + ab = 101ab + 10(a^2 + b^2)

Thus, we need to solve: (10a+b)(10b+a)=7272 (10a + b)(10b + a) = 7272 Calculating 7272 \sqrt{7272} gives an approximate range for ab ab and ba ba since the product is supposed to be around this number.

Step 4: Calculate Possible a a and b b

To find valid combinations of a a and b b , we can systematically check the options provided:

  1. Option 1: a=1,b=3 a = 1, b = 3

    • 10(1)+3=13 10(1) + 3 = 13
    • 10(3)+1=31 10(3) + 1 = 31
    • 13×31=403 13 \times 31 = 403 (not valid)
  2. Option 2: a=2,b=3 a = 2, b = 3

    • 10(2)+3=23 10(2) + 3 = 23
    • 10(3)+2=32 10(3) + 2 = 32
    • 23×32=736 23 \times 32 = 736 (not valid)
  3. Option 3: a=1,b=2 a = 1, b = 2

    • 10(1)+2=12 10(1) + 2 = 12
    • 10(2)+1=21 10(2) + 1 = 21
    • 12×21=252 12 \times 21 = 252 (not valid)
  4. Option 4: a=5,b=2 a = 5, b = 2

    • 10(5)+2=52 10(5) + 2 = 52
    • 10(2)+5=25 10(2) + 5 = 25
    • 52×25=1300 52 \times 25 = 1300 (not valid)

After checking these combinations, we realize none of the combinations yields 7272 as the result.

5. Verify and Summarize

Upon checking, we find that no provided combinations of a a and b b correctly reconstruct the number 7272 through the multiplicative expression ab×ba ab \times ba .

Final Answer

There are no valid values of a a and b b among the provided options that satisfy ab×ba=7272 ab \times ba = 7272 . Each combination was validated, yielding different products.

This problem has been solved

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