# Find this Unique UVC Palindrome ( ignoring signs and decimal) from Given Fractional Relationship

Given:

U, V, C are three distinct digits ( 0 to 9 ).

UVVVV and CVVVV.U are concatenated numbers.

Dot “.” Stands for decimal.

Relation:

$$UVVVV/C= CVVVV.U$$

Find U, V , C

• hmm inspiration for a mathematical puzzle with t, o and m or t, 0 and m maybe... :-)
– tom
May 19, 2019 at 16:31

## Finding $$C$$

1. If dividing an integer by $$C$$ gives a fraction with exactly one digit after the decimal point (note that $$U=0$$ doesn't work), then $$C$$ must be non-coprime with $$10$$, i.e. it must be one of $$2,4,5,6,8$$.

2. If $$C\geq45$$, then the right-hand side is more than $$40,000$$, and after multiplying by $$C$$ it won't be a 5-digit number any more. So we must have $$C=2$$.

## Finding $$U$$ and $$V$$

1. Since $$C=2$$, the division by $$C$$ must give $$U=5$$.

2. Since $$UVVVV$$ divided by $$2$$ is not an integer, $$V$$ must be odd. Trying the possibilities in turn shows that $$V=9$$ is the only one which works.

## Summary

$$U=5,V=9,C=2$$. The equation is $$59999/2=29999.5$$.

• C only has to be even (or 5) for part 1.
– JMP
May 19, 2019 at 15:58
• To be precise, the condition is that the prime factors of C contain 2 and/or 5. This makes the following numbers valid: 2,4,5,6,8. Point 2 can still get the right value with the same method.
– Leo
May 20, 2019 at 1:42
• @JonMarkPerry and Leo: Oops! Thanks for the tip; I modified my answer just slightly to take these possibilities into account. May 20, 2019 at 7:53
• you can use $\gcd(C,10)\gt1$ for 'non-coprime'
– JMP
May 20, 2019 at 8:03

$$59999/2=29999.5$$

because:

$$C=1,2,3$$ due to RHS being $$\sim C^2$$ in magnitude, which must be five digits. $$C=1$$ means $$U=0$$ which is impossible, and $$C=3$$ means $$(C\times .U) \pmod 1 \equiv 0$$ which is also impossible. Therefore $$C=2, U=5$$.

and then:

We now have$$\frac{5VVVV}{2}=2VVVV.5$$ which leads to $$5000+\frac{VVVV}{2}=VVVV.5$$ by cancelling $$20,000$$ from each side. So $$10000+VVVV=2VVVV+1$$ and then $$VVVV=9999$$, so $$V=9$$.

Ok so others have got there before me, but I have a slightly different way of approaching I think...

C cannot be 1,3,7,9,0 because.. 1 would not give a decimal point, 3 7 9 would give more than one decimal point or no decimal points e.g. 2/3 = 0.66666etc., and we cannot conveniently divide by 0 - thus C must be 2,4,5,6,8

now

C = U/C more or less because when the divide the first digit, U, by C we need to get C in the first column... Now for C=2 we could have U=4, but we can't have .4 at the end of the answer we need .5 so U must be 5 if C=2 - this works because 5VVVV/2 = 2VVVV.5 is possible if V is odd... but if we try C=4,5,6,8 we cannot find a single digit that will fit e.g. 8VVVV/4 = 2VVVV... and we need 4 at the front... thus for C>2 we cannot get the first digit to work - thus C=2 and U=5 (unless, of course, C=3 and U=9, but then we would not get the single digit after the decimal point at the end of number... )

finally

By inspection V=9 works if C=2 and U=5 and V=1,3,5,7 do not work (of course even V does not give .5 at the end) so we have $$59999/2=29999.5$$

and

this works for any number of 9s in the middle... 5/2 = 2.5; 59/2 = 29.5;....; 59999999999/2 = 29999999999.5 etc.

• $\frac{33}{6}=5.5$
– JMP
May 19, 2019 at 16:32
• @JonMarkPerry - oh rats... now I understand your comment above better... to the first answer -- answer edited to correct this
– tom
May 19, 2019 at 16:33
• @tom..sure..generating all infinite number of non-prime palindromes
– Uvc
May 19, 2019 at 16:43