2-digit header makes 5-digit number

The cube of a 2-digit integer, B R 3 \overline{BR} ^3 is equal to a 5-digit integer, B R A I N \overline{BRAIN} .

If B , R , A , I , N B,R,A,I,N are all distinct single digit integers, submit your answer as the 2-digit integer B R \overline{BR} .


The answer is 32.

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

Linkin Duck
Mar 24, 2017

From the equation provided we have:

B R × 1000 + 1000 > B R 3 B R × 1000. \overline{BR} \times 1000 + 1000 > \overline{BR} ^ 3 \geq \overline{BR } \times 1000 .

Since 10 B R 100 10 \leq \overline{BR} \leq 100 , dividing by B R \overline{BR} , we get

1000 + 100 1000 + 1000 B R > B R 2 1000. 1000 + 100 \geq 1000 + \frac{1000} { \overline{BR} } > \overline{BR}^2 \geq 1000.

Hence, 34 > B R > 31 34 > \overline{BR} > 31 . We now check the cases

  • B R = 33 B R 3 = 35937 \overline{BR} = 33 \Rightarrow \overline{BR} ^ 3= 35937 , which does not give us distinct digits.
  • B R = 32 B R 3 = 32768 \overline{BR} = 32 \Rightarrow \overline{BR} ^ 3= 32768 , which gives us distinct digits.

Hence, B R = 32 \overline{BR} = 32 .

BR=33 need not be checked as R has to be different from B. So 32 is the answer.

j chaturvedi - 4 years, 1 month ago
Desmond Yeung
Apr 10, 2017

Consider 10^3= 1000, 20^3=8000, 30^3=27000, 40^3= 64000. We can deduce that the number BR lies between 30 and 40, which means B=3

Trial and error: Try BR=35 (start from the middle 30s) 35^3=42875, which is too large. we repeat trying a smaller number and finally arrives at BR=32

Any odd multiple of 5, cubed, ends in 5. So you can rule out 35 too

Jonathan Drucker - 4 years, 2 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...