Technique

Least multiplier that makes a fraction a whole number

Arithmetic • Number System • LCM and HCF

Statement

Write the given amount as

\frac{p}{q}

with

g cd (p, q) = 1

. Then

n \times \frac{p}{q}

is an integer exactly when

q

divides

n

, so the least such positive

n

q

itself.

How to use it

1Reduce to lowest terms first — the unreduced denominator overstates the answer.
2For a decimal amount, clear the decimal point and then cancel: $0.36 = \frac{36}{100} = \frac{9}{25}$ , so $25$ is the least multiplier.
3Every multiplier that works is a multiple of $q$ , which answers 'how many such $n$ below a bound' as well as 'the least'.

What is the least number of identical items costing Rs.

2.25

each that can be bought for an exact number of rupees?

Step-by-step

1
$2.25 = \frac{225}{100} = \frac{9}{4}$
Why: Clear the decimal and cancel the common factor $25$ .
2
$g cd (9, 4) = 1 \Rightarrow 4 ∣ n$ , so $n_{m i n} = 4$
Why: Nothing in the numerator can help clear the denominator.

Answer

4

Watch out

The coprimality check is the whole step: if

p

and

q

share a factor the reduced denominator is smaller, and quoting the printed denominator gives a multiplier that works but is not the least.

Builds on

Also known as

smallest multiplier, make a fraction an integer, denominator in lowest terms, least number of days