Find the ways to express 11025 as product of two factors. (a) 13 (b) 14 (c) 26 (d) 27


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Question Number 206309 by BaliramKumar last updated on 11/Apr/24

$Find the ways to express 11025$ $as product of two factors .$ $(a) 13 (b) 14 (c) 26 (d) 27$
	Answered by TheHoneyCat last updated on 11/Apr/24
	$11025 =5×2205 =5^2 ×441 =3×5^2 ×147 =3^2 ×5^2 ×49 =3^2 ×5^2 ×7^2 From there, it is easy to count how many tuples (x,y) are such that xy=11025 This is 3×3×3=27 Let us now count how many of them are such that (x,y)=(y,x) x=3^a ×5^b ×7^c y=3^(2−a) ×5^(2−b) ×7^(2−b) ⇒a=b=c=1 so this is only one possibility. So there are 26 unsymetric tuples and 1 symetric leading to: 13+1 pairs {x,y} such that xy=11025 So overall: #{{x,y}∣xy=11025} = 14 _□$
	$11025$ $= 5 \times 2205$ $= 5^{2} \times 441$ $= 3 \times 5^{2} \times 147$ $= 3^{2} \times 5^{2} \times 49$ $= 3^{2} \times 5^{2} \times 7^{2}$ $From there, it is easy to count how many$ $tuples (x, y) are such that x y = 11025$ $This is 3 \times 3 \times 3 = 27$ $Let us now count how many of them$ $are such that (x, y) = (y, x)$ $x = 3^{a} \times 5^{b} \times 7^{c}$ $y = 3^{2 - a} \times 5^{2 - b} \times 7^{2 - b}$ $\Rightarrow a = b = c = 1 so this is only one possibility .$ $So there are 26 unsymetric tuples$ $and 1 symetric$ $leading to : 13 + 1 pairs {x, y} such that$ $x y = 11025$ $So overall :$ $You can't use 'macro parameter character #' in math mode$
	Commented by TheHoneyCat last updated on 11/Apr/24

	$By the way, for those who want to check,$ $here is the full list in the form$ $c, b, a ∣ x \times y ∣ n$ $with$ $x ⩽ y$ $x y = 11025$ $x = 3^{a} 5^{b} 7^{c}$ $n such that \forall x \exists! n$
	Commented by TheHoneyCat last updated on 11/Apr/24

	$000 1 \times 11025 1$ $001 3 \times 3675 2$ $002 9 \times 1125 5$ $010 5 \times 2205 3$ $011 15 \times 735 6$ $012 45 \times 245 10$ $020 25 \times 441 8$ $021 75 \times 147 13$ $022 225$ $100 7 \times 1575 4$ $101 21 \times 525 7$ $102 63 \times 175 12$ $110 35 \times 245 9$ $111 105 \times 105 14 \leftarrow Max Value$ $112 315$ $120 175$ $121 525$ $122 1575$ $200 49 \times 255 11$ $201 147$ $202 441$ $210 245$ $211 735$ $212 2205$ $220 11225$ $221 3675$ $222 1125$
	Commented by BaliramKumar last updated on 12/Apr/24

	$Thanks$
	Answered by mr W last updated on 11/Apr/24

	$11025 = 3^{2} 5^{2} 7^{2}$ $\Rightarrow n u m b e r o f f a c t o r s : 3 \times 3 \times 3 = 27$ $1 + \frac{27 - 1}{2} = 14$ $\Rightarrow 14 w a y s$
	Commented by BaliramKumar last updated on 12/Apr/24

	$Thanks$
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