Square any number ending in 5

Question

Solve in your head, no written work. What is 85^2? Aim: under 4 seconds. Then explain the shortcut.

Accepted Answer

Split off the tens digit. For 85 the tens digit is 8: multiply it by the next integer, 8 	imes 9 = 72, and stick 25 on the end: 85^2 = 7225. Why it works. Write the number as 10a + 5 (here a = 8). Then (10a + 5)^2 = 100a^2 + 100a + 25 = 100\,a(a + 1) + 25. The 100\,a(a+1) is "a(a+1) followed by two zeros," and the constant 25 fills those last two digits. So the answer is always the product a(a+1) written down, then 25 appended. Quick checks: 35^2 = (3 	imes 4)\,25 = 1225; 65^2 = (6 	imes 7)\,25 = 4225; 125^2 = (12 	imes 13)\,25 = 15625. Takeaway. For any number ending in 5, square it as a…

Square any number ending in 5

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