Are you smart enough to multiply two numbers

Number Theory Level 2

We have two, two digit numbers AB and CD if the value of A×C=20 , (A×D)+(B×C)=48 and B×D=16 calculate the value of AB×CD

Note:-Here AB and CD means two digit numbers not multiplication And A×C, A×D ,B×C and B×D means multiplication

Hint:-You have to do something with distributive property of multiplication under addition


The answer is 2496.

Aman Sharma by Aman Sharma , 25, India

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2 solutions

Gajula Chinmayi
Aug 13, 2014

AB = 10A+B CD = 10C+D AB×CD = (10A+B)(10C+D) = 100A×C + 10A×D + 10 B×C + B×D =100×20 + 10[(A×D)(B×C)] + 16 =2000+480+16 =2496

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You should still verify that such integers can exist. In this case, we get A B = 48 , C D = 52 AB = 48, CD = 52 .

Calvin Lin Staff - 6 years, 10 months ago

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You are motivation for me sir

Aman Sharma - 6 years, 10 months ago
Matthew Cox
Dec 16, 2014

For now, let's dispense with the idea that A, B, C, and D are digits in 2-digit numbers - let's just call them variables a, b, c, and d. We have that (1) ac=20, (2) ad+bc=48, and (3) bd=16. Multiply (1) and (3) to get that (4) abcd=320, which can be rearranged to ad * bc=320. When examined in conjunction with (2), ab and cd are 40 and 8, but we don't know in which order. We can note that, from (1) and (3), 20 can be broken up into 4 * 5, and 16 can be 2 * 8. Also note that 40 can be 5 * 8, and 8 can be 2 * 4. With some rearrangement and re-checking, we discover that a=5, b=2, c=4, and d=8. Then, it's a simple matter of evaluating 52 * 48, which is 2496.

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