Multiply 63 × 67 with the "units add to 10" trick

Question

When two numbers share a tens digit and their units add to 10, the product splits into two effortless halves. Solve in your head. What is 63 	imes 67? Aim: under 10 seconds.

Accepted Answer

Tens digit is 6, and units 3 + 7 = 10. Multiply the tens digit by the next integer for the front, and multiply the units for the back: 6 	imes 7 = 42, \qquad 3 	imes 7 = 21 \ \Rightarrow\ 63 	imes 67 = 4221. So 63 	imes 67 = \mathbf{4221}. Check: 63 	imes 67 = 63 	imes 70 - 63 	imes 3 = 4410 - 189 = 4221. Matches. Why: (10a+b)(10a+c) with b+c=10 equals 100\,a(a+1) + bc. Takeaway. Same tens digit, units adding to 10: front = a(a+1), back = (	ext{units product}). Perfect for 63×67, 24×26, 81×89.

Multiply 63 × 67 with the "units add to 10" trick

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