SSC CGL Maths · Number System

Division Algorithm and Multiples

Every division gives a quotient and a remainder smaller than the divisor, and the remainder tells you exactly how far a number is from the nearest multiples.

All Number System notes

Why this matters

Eight questions in the bank, seven EASY and one MODERATE. They all use one equation, dividend equals divisor times quotient plus remainder: forwards to rebuild a dividend, and backwards to find the nearest multiple of a number or the least amount to add or take away.

Concept 1 of 2: The division algorithm

Share 47 sweets among 5 children: each gets 9 and 2 are left over. The sweets you started with are the shares plus the leftover, and the leftover is always fewer than the number of children, or each child could have had one more.

Definition

Dividend $=$ divisor $\times$ quotient $+$ remainder, and the remainder is at least 0 and less than the divisor.
'Exactly divisible' means the remainder is 0.
When a stem links the divisor to the quotient and the remainder ('the divisor is 6 times the remainder'), start from the one number it gives you directly and follow the links one at a time.
Fast route for an exact division where the dividend and the divisor are both changed by a percentage: the new quotient is the old one times the dividend's factor over the divisor's factor. If the dividend is raised by 20% and the divisor cut by 25%, the quotient becomes $q \times \frac{1.2}{0.75} = 1.6q$ .

Division algorithm

\text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder}, \qquad 0 \le \text{Remainder} < \text{Divisor}

Worked example

In a division, the divisor is 5 times the remainder and the quotient is 3 more than the divisor. If the remainder is 7, find the dividend.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 11 Sep 2024, Shift 1 · Q73Easy

Example 1 · Number System · Division Algorithm and Multiples

In a division problem, the divisor is 4 times the quotient and 2 times the remainder. If the remainder is 32, then find the dividend.

Accepting a remainder as large as the divisor

A remainder is always less than the divisor. If your working gives a remainder equal to or bigger than the divisor, the quotient is too small.

Scaling the quotient of a division that had a remainder

Multiplying the quotient by the ratio of the two factors is exact only when the original division left remainder 0. Otherwise rebuild the dividend and divide again.

Drill 2 more on the division algorithm

Concept 2 of 2: The nearest multiple of a number

Divide and look at the remainder. It is how far you are above the multiple just below; the divisor minus the remainder is how far you are below the multiple just above.

Definition

Divide: $N = dq + r$ .
Largest multiple of $d$ not above $N$ : $N - r$ . So the least number to subtract is $r$ .
Smallest multiple of $d$ not below $N$ : $N + (d - r)$ when $r \ne 0$ . So the least number to add is $d - r$ .
Largest $n$ -digit multiple: take $N = 99\ldots9$ (the largest $n$ -digit number) and subtract the remainder.
Smallest $n$ -digit multiple: take $N = 10\ldots0$ (the smallest $n$ -digit number) and add $d - r$ .
Fast route: you rarely need the full division. Jump to a round multiple first ( $53 \times 200 = 10600$ ), then take away or add small multiples of $d$ until you reach $N$ .

Distance to the nearest multiples

N = dq + r: \quad \text{subtract } r, \ \text{or add } d - r

Nthe given number
dthe divisor
rthe remainder when N is divided by d

Worked example

Find the smallest 4-digit number and the largest 4-digit number that are exactly divisible by 37.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 10 Sep 2024, Shift 1 · Q56Easy

Example 2 · Number System · Division Algorithm and Multiples

Which of the following is the largest 5-digit number divisible by 47?

Adding the remainder instead of the gap

To reach the next multiple you add the divisor minus the remainder. Adding the remainder itself overshoots or falls short: with

650 = 30 \times 21 + 20

, you add 10, not 20.

Landing on a multiple with the wrong number of digits

For the smallest 5-digit multiple, start from 10000, not from 9999. A multiple just below 10000 has only four digits, so check the digit count of your answer before you mark it.

Drill 4 more on the nearest multiple of a number

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

The division algorithm
Division algorithm
$\text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder}, \qquad 0 \le \text{Remainder} < \text{Divisor}$
The nearest multiple of a number
Distance to the nearest multiples
$N = dq + r: \quad \text{subtract } r, \ \text{or add } d - r$

Watch out for (4)

Accepting a remainder as large as the divisor
→ The division algorithm
Scaling the quotient of a division that had a remainder
→ The division algorithm
Adding the remainder instead of the gap
→ The nearest multiple of a number
Landing on a multiple with the wrong number of digits
→ The nearest multiple of a number

Test yourself on Number System

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