How many zeroes at the end of $3^34^45^56^6 - 3^64^55^46^3$?

The quantity $3^34^45^56^6 - 3^64^55^46^3$ will end in how many zeros?

This is a GRE Practice question and I have to do it without using a calculator. Can anyone help me on this?

• It is a good thing that it is without calculator, the GRE has lots of questions, and a calculator would slow one down. – André Nicolas Jul 28 '13 at 18:06

Using $a^m\cdot a^n=a^{m+n}; a^{mn}=(a^m)^n$ and $(a\cdot b)^m=a^m\cdot b^m$ where $a,b,m$ are real numbers

$$3^34^45^56^6 - 3^64^55^46^3$$

$$=3^3(2^2)^45^5(2\cdot3)^6 - 3^6(2^2)^55^4(2\cdot3)^3$$

$$= 2^{8+6}5^53^{6+3} - 3^{3+6}2^{10+3}5^4$$

$$=2^{13}5^43^9(2\cdot5-1)$$

As $10=2\cdot5,$ the highest power of $10$ will be min $(13,4)$

Step 1: Factor and simplify.

Step 2: Only powers of 2, 5 are relevant in answering the question.

Note that $$3^34^45^56^6-3^64^55^46^3=5^4(3^34^45^16^6-3^64^56^3)$$

There are always enough factors of $2$ to match the powers of $5$ to give at least four zeros - but the term in brackets cannot be by $5$. This is because $5$ divides the first product in the brackets, but not the second, so cannot divide their difference.

S = $3^34^45^56^6 - 3^64^55^46^3$

=$\underbrace{3^32^36^6(5*2)^5}_{\mathcal A} - \underbrace{ 3^62^66^3(5*2)^4}_{\mathcal B}$

So we see above that both A and B have ( unequal number of )trailing zeroes. A has 5 Zeroes while B has 4 at its end.
In this case for S= A-B...

number of zeroes at the end of S will be the number of zeroes at the end of of the lesser of the two numbers amongst A and B viz 4

as the fifth digit of S is going to be non zero ( substraction of a non zero digit of B from zero; the fifth trailing zero of A)

EDIT: In response to the query by OP; the logic remains same in case of addition so A+B again will have 4 zeros

• You could have made this shorter by just saying "$3^3 4^4 5^5 6^6$ has $5$ zeroes while $3^6 4^5 5^4 6^3$ has $4$ zeroes so their difference has $4$ zeroes" – Henry Jul 28 '13 at 21:38
• That would be fast enough for GRE – ARi Jul 29 '13 at 11:03
• @ARi What if the question involved addition instead of subtraction of the two terms? Then would the answer be 5? – user87274 Jul 31 '13 at 0:06
• It will still be 4 – ARi Jul 31 '13 at 12:58
• In fact this is the shortest solution – ARi Aug 2 '13 at 15:31