Two positive integers m and n satisfy HCF (m, n) = 14 and LCM (m, n) = 168. If m < n and m divides 84, what is the maximum possible value of m?

Using the relationship between HCF and LCM: $HCF \times LCM = m \times n$

$14 \times 168 = m \times n$

$m \times n = 2352$

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Since $HCF(m, n) = 14$, both $m$ and $n$ must be multiples of 14.

We can write: $m = 14a$ and $n = 14b$, where $a$ and $b$ have no common factors (their HCF is 1).

Substituting in the product:

$14a \times 14b = 2352$

$196ab = 2352$

$ab = 12$

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Finding pairs $(a, b)$ where $ab = 12$ and $HCF(a, b) = 1$:

$a$	$b$	$m = 14a$	$n = 14b$
1	12	14	168
3	4	42	56
4	3	56	42
12	1	168	14

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Applying the condition $m < n$, valid pairs are:

$m$	$n$
14	168
42	56

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Checking which values of $m$ divide 84:

• Does 14 divide 84? Yes → $84 = 14 \times 6$ ✓
• Does 42 divide 84? Yes → $84 = 42 \times 2$ ✓

Both 14 and 42 satisfy all conditions. The maximum value of $m$ is 42.