CSAT Solved Papers/ 2025/Q35

2025 CSAT — Q35

Quant Number theory 2.5 marks Medium

What is the maximum value of $n$ such that $7 \times 343 \times 385 \times 1000 \times 2401 \times 77777$ is divisible by $35^n$ ?

A 3
B 4 Answer
C 5
D 7

Worked rationale

$35 = 5 \times 7$ , so $35^n$ needs $n$ fives and $n$ sevens. The answer is $\min(\nu_5, \nu_7)$ over the whole product, where $\nu_p$ is the total power of $p$ .

Factor each term, tracking only $5$ s and $7$ s:

term	$\nu_7$	$\nu_5$
$7$	$1$	$0$
$343 = 7^3$	$3$	$0$
$385 = 5\cdot7\cdot11$	$1$	$1$
$1000 = 2^3 5^3$	$0$	$3$
$2401 = 7^4$	$4$	$0$
$77777 = 7 \cdot 11111$	$1$	$0$
total	$\mathbf{10}$	$\mathbf{4}$

(Note $77777 = 7 \times 11111 = 7 \times 41 \times 271$ — one factor of $7$ , no $5$ .)

\nu_5 = 4,\quad \nu_7 = 10 \ \Rightarrow\ n = \min(4, 10) = 4.

The supply of $5$ s is the bottleneck.

Answer: (b) 4.

Why the other options miss

A

miscounted a repeated factor: counts the $5$ s only in $1000$ ( $5^3$ ) and misses the single $5$ hidden in $385$ , under-counting $\nu_5$ to $3$ .
C

an arithmetic slip: mis-adds the $5$ s (e.g. reads $1000 = 5^3$ plus a phantom extra) or averages the two valuations.
D

wrong formula: counts only $\nu_7$ -like quantities, or takes the max instead of the min of the two prime supplies, ignoring that $35^n$ needs both.

Specialist insight

For “highest power of a composite dividing a product,” factor the divisor into primes and take the minimum of the per-prime supplies — never the maximum, never the sum. Here $7$ is abundant ( $\nu_7 = 10$ ) but $5$ is scarce ( $\nu_5 = 4$ ), so $5$ caps $n$ at $4$ . The deadliest slip is missing the $5$ camouflaged inside $385 = 5\cdot7\cdot11$ (the obvious source is $1000$ , but $385$ quietly adds one). The discipline: factor every term fully, sum each prime separately, take the min. Watch for the “ugly” factor ( $77777$ ) designed to look like it might add a $5$ — it does not.

The trap, in one line

$35^n$ needs $n$ fives *and* $n$ sevens, so $n = \min(\nu_5,\nu_7) = \min(4,10) = 4$ — the $5$ s (don't miss the one in $385$ ) are the bottleneck.

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