2024 CSAT — Q9

Quant Number theory 2.5 marks Hard

How many consecutive zeros are there at the end of the integer obtained in the product $1^2 \times 2^4 \times 3^6 \times 4^8 \times \cdots \times 25^{50}$ ?

A 50
B 55
C 100
D 200 Answer

Worked rationale

Trailing zeros = the number of times $10$ divides the product = $\min(\nu_2, \nu_5)$ , where $\nu_p$ is the total exponent of prime $p$ . In any product built from $1,2,\dots,25$ the supply of $2$ s dwarfs the supply of $5$ s, so the answer is governed entirely by $\nu_5$ — do not waste clock on the $2$ s.

Read the pattern: the $n$ -th factor is $n^{2n}$ (term $n$ raised to exponent $2n$ ). Only the multiples of $5$ contribute $5$ s:

base $n$	exponent $2n$	fives in the base	fives contributed
$5$	$10$	$1$	$10$
$10$	$20$	$1$	$20$
$15$	$30$	$1$	$30$
$20$	$40$	$1$	$40$
$25$	$50$	$2$ ( $25=5^2$ )	$100$

\nu_5 = 10+20+30+40+100 = 200.

Since $\nu_2 \gg 200$ (the even bases alone pour in thousands of $2$ s), $\min(\nu_2,\nu_5)=200$ .

Answer: (d) 200.

Why the other options miss

A

an arithmetic slip that also misses a case: counts only the single base $25$ (exponent $50$ ) or stops after one term, ignoring that 5, 10, 15, 20 each contribute too.
B

answered the sub-step, not the question: counts one $5$ from each multiple-of-5 base ( $5,10,15,20$ ) plus $25$ as two, i.e. $1+1+1+1+2 = 6$ … then mis-multiplies, or counts bases ( $5{+}10{+}15{+}20{+}25 = 75$ ) and mis-corrects. It is the “looks like I did something with the exponents” trap — close enough to feel earned, wrong because the exponents must be summed, not the bases.
C

missed a case: the single deadliest near-miss. Counts the $5$ s from $5,10,15,20$ correctly ( $10+20+30+40=100$ ) but forgets that $25=5^2$ doubles its contribution — or, conversely, counts only the $25^{50}=5^{100}$ block and forgets $5,10,15,20$ . Either way you land on exactly half.

Specialist insight

This is a $\nu_5$ -only problem in disguise — the moment you see “trailing zeros of a product of small numbers,” stop computing and write only the multiples of $5$ . The whole template collapses to: sum the exponents on each multiple of 5, then add one extra hit for every base divisible by $25$ (two extra for $125$ , etc.). The exam plants its trap on the $5^2$ inside $25$ — every CSAT trailing-zeros item that includes $25$ , $50$ , $100$ , or $125$ is testing whether you remembered the second power. 10-second sanity check: the largest answer that only counts first powers is $100$ ; the real answer must exceed it, so $200$ is structurally the only candidate above $100$ — you can almost key it without finishing.

The trap, in one line

Forgetting that $25 = 5^2$ contributes its $5$ twice — the difference between (c) 100 and (d) 200.

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