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# Bases, Exponents and Role reversals (I.S.I. Entrance 2018)

[et_pb_section fb_built="1" admin_label="Blog Hero" _builder_version="3.22" use_background_color_gradient="on" background_color_gradient_start="rgba(114,114,255,0.24)" background_color_gradient_end="#ffffff" background_blend="multiply" custom_padding="0|0px|0|0px|false|false" animation_style="slide" animation_direction="top" animation_intensity_slide="2%" locked="off"][et_pb_row _builder_version="3.25" background_size="initial" background_position="top_left" background_repeat="repeat" custom_margin="|||" custom_padding="27px|0px|27px|0px" custom_width_px="1280px"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_text _builder_version="3.27.4" text_text_color="#474ab6" text_line_height="1.9em" background_size="initial" background_position="top_left" background_repeat="repeat" text_orientation="center" max_width="540px" module_alignment="center" locked="off"]Let $a, b, c$ are natural numbers such that $a^{2}+b^{2}=c^{2}$ and $c-b=1$. Prove that
(i) a is odd.
(ii) b is divisible by 4
(iii) $a^{b}+b^{a}$ is divisible by c
[/et_pb_text][/et_pb_column][/et_pb_row][/et_pb_section][et_pb_section fb_built="1" admin_label="Blog" _builder_version="3.22" custom_margin="|||" custom_padding="0px|0px|21px|0px|false|false"][et_pb_row _builder_version="3.25" background_size="initial" background_position="top_left" background_repeat="repeat" max_width="960px" custom_padding="0|0px|24px|0px|false|false" use_custom_width="on" custom_width_px="960px"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_tabs _builder_version="3.12.2"][et_pb_tab title="Hint 1 - Isolate a" _builder_version="3.12.2"]Notice that $a^2 = c^2 - b^2 = (c+b)(c-b)$

But c - b = 1. Hence $a^2 = c + b$. But c and b are consecutive numbers (after all their difference is 1!). Sum of two consecutive numbers is always odd (Why?).

Hence $a^2$ is odd. This implies a is odd.
[/et_pb_tab][et_pb_tab title="Hint 2 - Eliminate c" _builder_version="3.12.2"]Replace c by 1+b. We have $a^2 + b^2 = ( 1+ b)^2 = 1 + b^2 + 2b$

But that means $a^2 = 1 + 2b$. This implies $2b = a^2 - 1 = (a-1)(a+1)$

We already showed a is odd. Hence odd + 1, odd -1 are consecutive even numbers. Hence atleast one of them must be divisible by 4 implying their product must be divisible by 8 (as the other is divisible by at least 2).

Hence 2b equals something that is divisible by 8. This implies b is divisible by 4.
[/et_pb_tab][et_pb_tab title="Hint 3 - A bit of Modular Arithmetic" _builder_version="3.12.2"]We already found that $a^2 = c + b$. We test this modulo c (Modular Arithmetic is a useful tool from Number Theory that you should definitely learn).

$a^2 \equiv c + b \equiv b \mod c$ . But since b = c - 1, hence $b \equiv -1 \mod c$. Hence $a^2 \equiv -1 \mod c$

Now recall that b is divisible by 8. Hence b/2 is even. This implies $(a^2)^{\frac{b}{2}} \equiv (-1)^{\frac{b}{2}} = 1 \mod c$. Hence $a^b \equiv 1 \mod c$

On the other hand a is odd and b = c-1. Hence $b^a \equiv (-1)^a \equiv -1 \mod c$

Adding we get the final answer $a^b + b^a \equiv 1 - 1 \equiv 0 \mod c$
[/et_pb_tab][/et_pb_tabs][/et_pb_column][/et_pb_row][/et_pb_section][et_pb_section fb_built="1" admin_label="Footer" _builder_version="3.22" background_color="#f7f8fc" custom_padding="0px|0px|2px|0px|false|false" animation_style="zoom" animation_direction="bottom" animation_intensity_zoom="6%" animation_starting_opacity="100%" saved_tabs="all"][et_pb_row use_custom_gutter="on" gutter_width="2" _builder_version="3.25" background_size="initial" background_position="top_left" background_repeat="repeat" custom_padding="24px|0px|145px|0px|false|false" column_structure="1_2,1_4,1_4"][et_pb_column type="1_2" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_text _builder_version="3.27.4" text_text_color="#7272ff" header_font="|on|||" header_text_color="#7272ff" header_font_size="36px" header_line_height="1.5em" background_size="initial" background_position="top_left" background_repeat="repeat" custom_margin="||20px|" animation_style="slide" animation_direction="bottom" animation_intensity_slide="10%"]

## Get Started with I.S.I. Entrance Program

[/et_pb_text][et_pb_text _builder_version="3.27.4" text_text_color="#8585bd" text_font_size="22px" text_line_height="1.9em" background_size="initial" background_position="top_left" background_repeat="repeat" animation_style="fade" locked="off"]Outstanding mathematics for brilliant school students.