if the fraction ((m^2 +25m)/(m+1)) is reductible. how many values does m take if is a 2 digit number? thanks


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Question Number 215999 by lmcp1203 last updated on 25/Jan/25
$if the fraction ((m^2 +25m)/(m+1)) is reductible. how many values does m take if is a 2 digit number? thanks$
$i f t h e f r a c t i o n \frac{m^{2} + 25 m}{m + 1} i s r e d u c t i b l e . h o w m a n y v a l u e s d o e s m t a k e i f i s a 2 d i g i t n u m b e r ? t h a n k s$
	Answered by Rasheed.Sindhi last updated on 25/Jan/25
	$((m^2 +25m)/(m+1))=m(((m+25)/(m+1))) =m(((m+1+24)/(m+1)))=m(1+((24)/(m+1))) ((m^2 +25m)/(m+1)) is reducible⇒((24)/(m+1)) is reducible. 24=2^3 .3 m+1=2p,3q,6r m=2p−1,3q−1,6r−1 ∧ 10≤m≤99 { ((10≤2p−1≤99⇒6≤p≤50)),((10≤3q−1≤99⇒4≤q≤33)),((10≤6r−1≤99⇒2≤r≤16)) :} m=2p−1: 6≤p≤50 : 45 values m=3q−1: 4≤q≤33 : 30 values m=6r−1: 2≤r≤16 : 15 values common Total 45+30−15=60 values$
	$\frac{m^{2} + 25 m}{m + 1} = m (\frac{m + 25}{m + 1})$ $= m (\frac{m + 1 + 24}{m + 1}) = m (1 + \frac{24}{m + 1})$ $\frac{m^{2} + 25 m}{m + 1} i s r e d u c i b l e \Rightarrow \frac{24}{m + 1} i s r e d u c i b l e .$ $24 = 2^{3} . 3$ $m + 1 = 2 p, 3 q, 6 r$ $m = 2 p - 1, 3 q - 1, 6 r - 1 \land 10 ⩽ m ⩽ 99$ ${\begin{cases} 10 ⩽ 2 p - 1 ⩽ 99 \Rightarrow 6 ⩽ p ⩽ 50 \\ 10 ⩽ 3 q - 1 ⩽ 99 \Rightarrow 4 ⩽ q ⩽ 33 \\ 10 ⩽ 6 r - 1 ⩽ 99 \Rightarrow 2 ⩽ r ⩽ 16 \end{cases}$ $m = 2 p - 1 : 6 ⩽ p ⩽ 50 : 45 v a l u e s$ $m = 3 q - 1 : 4 ⩽ q ⩽ 33 : 30 v a l u e s$ $m = 6 r - 1 : 2 ⩽ r ⩽ 16 : 15 v a l u e s c o m m o n$ $T o t a l 45 + 30 - 15 = 60 v a l u e s$
	Commented by lmcp1203 last updated on 25/Jan/25

	$t h a n k y o u$
	Answered by A5T last updated on 25/Jan/25

	$\frac{m^{2} + 25 m}{m + 1} = m + 24 - \frac{24}{m + 1}$ $\Rightarrow \gcd (m^{2} + 25 m, m + 1) = \gcd (24, m + 1)$ $\frac{m^{2} + 25 m}{m + 1} is reducible when (24, m + 1) \neq 1$ $That is when m + 1 contains either 3 or 2 as a$ $prime factor$ $Cardinality of such numbers :$ $⌊ \frac{99}{3} ⌋ + ⌊ \frac{99}{2} ⌋ - ⌊ \frac{99}{6} ⌋ - (⌊ \frac{9}{3} ⌋ + ⌊ \frac{9}{2} ⌋ - ⌊ \frac{9}{6} ⌋)$ $= 33 + 49 - 16 - 3 - 4 + 1 = 60$
	Commented by lmcp1203 last updated on 25/Jan/25

	$t h a n k y o u$
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