Study Guide · Chapter 34
12. Common Mistakes Aspirants Make
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- Confusing “divisible by A and B” with “divisible by A×B.” Correct approach: check the LCM of A and B, not the raw product, unless A and B are co-prime (e.g., 4 & 6 → use 12, not 24).
- Applying the digit-sum rule to divisibility by 6, 12, 18, etc. directly. These are composite divisors that must be broken into co-prime factors (e.g., 6 = 2×3, not tested by any single digit-sum trick of its own).
- Forgetting that the divisibility-by-11 rule uses alternating sum, not plain digit sum. Many aspirants add all digits like the rule for 3/9 and get it wrong.
- Sign errors in the alternating sum for 11: Losing track of which digits are “odd position” vs “even position” (always count from the right, starting at position 1 = units digit).
- Stopping the osculator process too early for 7/13/17/19 and misreading a non-zero intermediate result as “not divisible” without reducing it fully to a number small enough to judge by inspection or a further known table of multiples.
- Forgetting to add BOTH ⌊ n/p⌋ AND ⌊ n/p^2⌋, etc., in Legendre’s Formula — stopping after the first term underestimates the power/trailing zero count (a very common error for n≥25 when finding zeros, since 25, 50, 75, 100 each contribute an extra factor of 5).
- Assuming every factorial has a distinct trailing-zero count. In reality, consecutive values of n often give the same zero count (e.g., 96!, 97!, 98!, 99! all have 22 zeros) — so “no factorial ends in exactly 23 zeros” type facts trip up aspirants who assume the count increases by exactly 1 each time.
- Misapplying a^n ± b^n rules without checking the parity of n. E.g., wrongly assuming (a+b) always divides a^n - b^n regardless of whether n is odd or even.
- Sign/parity confusion with an+bn and (a-b): There is no standard guaranteed divisibility of an+bn by (a-b) — don’t invent a rule here under exam pressure.
- Treating “0” as a non-multiple in the last-two/three/four-digit rules. E.g., last two digits “00” should be treated as divisible by 4, 25, etc. (since 0÷4=0 remainder 0).
- Arithmetic slips while computing digit sums of very large numbers under time pressure — always double-check by re-adding, since a single misread digit changes the entire answer.
- Assuming the highest power of a composite number c=p^a× q^b dividing n! is simply E_p(n!)+E_q(n!) or the smaller of the two raw exponent counts without first dividing by the required power a or b — e.g., for 12=22×31, you must compare ⌊ E_2(n!)/2⌋ with E_3(n!), not E_2(n!) directly with E_3(n!).
- Over-generalizing the “n consecutive integers divisible by n!” rule to assume a larger factorial also always divides — the guarantee is exactly n!, not any multiple of it (Section 8, Example 6).
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