Numbers and divisibility: factors, multiples and remainders

Why number facts pay off without a calculator

Bocconi describes quantitative reasoning as numerical and logical problems using basic mathematics, and calculators are not allowed on the test. That makes number structure worth knowing well: if you can see that a number is divisible by 9, or that a remainder repeats, you skip long division entirely. This lesson collects the facts that do the most work.

Divisibility from the digits

Divisible by	Test
2	last digit even
3	sum of digits divisible by 3
4	last two digits form a multiple of 4
5	last digit 0 or 5
6	divisible by 2 and by 3
8	last three digits form a multiple of 8
9	sum of digits divisible by 9
11	alternating sum of digits (first minus second plus third…) divisible by 11

Take 7,452. Its digits add to $7 + 4 + 5 + 2 = 18$, so it is divisible by 9 (and so by 3). Its last two digits, 52, are a multiple of 4. Because 4 and 9 share no factor, 7,452 is divisible by $4 \times 9 = 36$; indeed $36 \times 207 = 7452$. The alternating sum $7 - 4 + 5 - 2 = 6$ is not a multiple of 11, so 11 does not divide it.

The step "4 and 9 share no factor" matters. A number divisible by 4 and by 6 need not be divisible by 24: 12 is a counterexample. Combine tests only for numbers with no common factor, or use the LCM (below).

Prime factorisation is the master key

Every whole number greater than 1 is a product of primes in exactly one way. Split off small primes until nothing is left:

$$360 = 2 \times 180 = 2^2 \times 90 = 2^3 \times 45 = 2^3 \times 3^2 \times 5.$$

Remember that 1 is not prime and that 2 is the only even prime. Once a number is factorised, several questions become counting:

• Number of positive divisors. If $n = p^a q^b r^c$, a divisor picks each prime's power independently, from 0 up to its exponent, so there are $(a+1)(b+1)(c+1)$ divisors.
• Perfect squares. A number is a perfect square exactly when every exponent in its factorisation is even.
• Divisibility. $m$ divides $n$ when every prime power in $m$ also appears, at least as high, in $n$.

GCD and LCM

The greatest common divisor takes each prime the two numbers share, at the lower power. The least common multiple takes every prime that appears in either, at the higher power. For $84 = 2^2 \times 3 \times 7$ and $120 = 2^3 \times 3 \times 5$:

$$\gcd = 2^2 \times 3 = 12, \qquad \text{lcm} = 2^3 \times 3 \times 5 \times 7 = 840.$$

For two numbers, $\gcd \times \text{lcm}$ equals their product: $12 \times 840 = 10\,080 = 84 \times 120$, a quick check on your work.

Word problems signal which one they want. "The largest equal groups", "the biggest tile that fits exactly" or "share without anything left over" ask for a GCD. "When will both happen together again" or "the smallest number divisible by both" ask for an LCM.

Remainders

Dividing $n$ by $d$ gives $n = qd + r$ with $0 \le r < d$. A remainder is never negative and is always smaller than the divisor. The key fact for questions without a specific $n$: in sums and products you may replace each number by its remainder, then reduce again at the end. If $n$ leaves remainder 4 on division by 6, then $5n$ leaves the same remainder as $5 \times 4 = 20$, which is 2. The last digit of a number is its remainder on division by 10, which is why last-digit questions are remainder questions in disguise.

When a question describes $n$ only through a remainder, the fastest check is to pick the smallest number that fits, then a second one, and confirm that both give the same result.

Odd, even and signs

Odd plus odd is even; odd times odd is odd; any product with an even factor is even. A product of numbers is negative when an odd number of them are negative. When a question says "number" rather than "integer", remember that 0, negatives and fractions are allowed, and test them before you trust a pattern.