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

1. Break Down the Problem

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

2. Relevant Concepts

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

3. Analysis and Detail

Let's denote:

• $ab = 10a + b$
• $ba = 10b + a$

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

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

Step 4: Calculate Possible $a$ and $b$

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

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

   • $10(1) + 3 = 13$
   • $10(3) + 1 = 31$
   • $13 \times 31 = 403$ (not valid)

2. Option 2: $a = 2, b = 3$

   • $10(2) + 3 = 23$
   • $10(3) + 2 = 32$
   • $23 \times 32 = 736$ (not valid)

3. Option 3: $a = 1, b = 2$

   • $10(1) + 2 = 12$
   • $10(2) + 1 = 21$
   • $12 \times 21 = 252$ (not valid)

4. Option 4: $a = 5, b = 2$

   • $10(5) + 2 = 52$
   • $10(2) + 5 = 25$
   • $52 \times 25 = 1300$ (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 $a$ and $b$ correctly reconstruct the number 7272 through the multiplicative expression $ab \times ba$.

Final Answer

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