Week 2 · PREDICTABLE LOGIC · 10–15 hrs

Number properties, CR assumption & DS

Number properties: parity, primes, factors

Quantitative

Parity on GMAT Focus is rarely about a single integer. It is about expressions: whether $n^2+n$ is even, whether $p+q$ can be odd, whether a product of consecutive integers is even. The structural fact to run first: adding or subtracting any even number preserves parity, so strip the even terms and look at what remains. $n(n+1)$ is always even, and so is $n^2-n$.

Primes have one fact doing most of the work: 2 is the only even prime. When a stem pairs "prime" with a parity condition, it is usually forcing one variable to 2. Treat "odd prime" as a deliberate restriction — the test writes it because 2 would otherwise be live.

For factors, hold this: if $n=p_1^{a_1}{p}_2^{a_2}\cdots$, the number of positive divisors of $n$ is $(a_1+1)(a_2+1)\cdots$. The count is odd exactly when $n$ is a perfect square. The same fact settles "how many pairs $(a,b)$ with $ab=n$": half the count, rounded up — and when $n$ is a square the middle pair contributes one.

The standard way to lose points is listing divisors by hand. You forget $n$ itself, or you stop at $\sqrt{n}$ and double without noticing that a square has already been counted once.

The decisions that separate the answer from the trap:

• Whether $n$'s value matters or only its shape. Many stems ask about form; computing wastes time and invites slips.
• Whether the variables may be 0, 1, or negative. Testing only 2 and 3 is the common failure; 1 breaks most prime assumptions, 0 breaks most positive ones.
• Whether "prime" was intended to exclude 2.

Divisibility and remainders

Quantitative

The exam rarely asks you to divide. It asks you to bound an unknown integer or to decide whether a divisibility condition is pinned down at all.

Nearly every remainder problem reduces to one identity: $N=dq+r$, where $q$ is an integer and $0\le r<d$. The inequality does the work, not the equation. "$N$ leaves remainder 3 when divided by 7" plus "$N<40$" leaves a handful of candidates; a quotient named in the stem often exists only to be bounded.

Expect disguises. Trailing zeros of $n!$ (count factors of 5 — 2s are never scarce). "How many multiples of 3 lie between $a$ and $b$" (a difference of floors). Parity questions, which are remainders mod 2. DS stems asking "is $N$ divisible by 12", where you need 4 and 3 independently — divisibility by 6 and by 2 gets you nothing.

Two decisions separate the right answer from the tempting one. First, whether a statement constrains the modulus you need: "$N\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}4)$" says nothing about $N<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>8$. Second, whether arithmetic on remainders is legitimate — you may add and multiply them under the same modulus, not under a different one.

The mechanical way strong candidates drop the point: finding $N\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}3)$ and then treating 2 as the value of $N$.

Worth holding: for coprime $a$ and $b$, the residues of $N$ mod $a$ and mod $b$ determine $N$ mod $ab$.

Critical Reasoning: assumptions and the negation test

Verbal

The word "assumption" may never appear. GMAT Focus asks what the argument depends on, requires, or presupposes, and hides the same demand inside boldface and evaluate stems. The correct answer is necessary, not merely helpful — and the tempting wrong answer is precisely the statement that would strengthen the argument if true but that the argument can survive without.

Two decisions separate right from tempting. Necessary versus sufficient: reject anything the conclusion stands without. Linkage: the assumption must connect these premises to this conclusion, not to a broader claim the author never made.

The negation test settles the first decision mechanically. Negate the logical force, not the sentence:

• "All A are B" → "at least one A is not B"
• "Some A are B" → "no A is B"
• "No A are B" → "at least one A is B"

If the negated statement leaves the conclusion standing, discard the choice; if it destroys the argument, keep it. Suppose a sales jump followed an ad campaign. The needed assumption is not "many people saw the ads" but "nothing else caused the jump" — negate the latter and the conclusion collapses.

Where strong candidates bleed points: they negate, then argue with the negation instead of asking whether the conclusion still follows; and under time pressure they pick the strongest-sounding statement. Negate every finalist, including the one you like.

Data Sufficiency: the sufficiency judgement

Data Insights

Data Sufficiency does not test whether you can find an answer; it tests whether the information fixes one. Every question reduces to: across all cases the statement permits, is the answer to the question asked always the same? Sufficiency is not truth — "Is $x$ negative?" answered with a definitive no is sufficient.

The disguises are where statements look decisive but are not. A statement that fixes $x+y$ leaves $x$ alone. A word problem about people or objects silently requires an integer. A geometry statement fixing two sides of a triangle does not fix the third. A statement giving $x^2=16$ looks like it yields a value; for "What is $x$?" it does not.

Three decisions separate the right answer from the tempting one:

• Value questions: one working case proves nothing. Two valid cases with different answers prove insufficiency — this is your entire tool for eliminating.
• Erase statement 1 before judging statement 2. Carrying a value across is the most common mechanical way strong candidates lose points here.
• Two solutions for the variable can still be sufficient if both produce the same answer to the question actually asked.

Have this in memory: a statement is sufficient exactly when the set of permitted cases maps to a single answer to the question posed — no more, no less.

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