If a % b = a² - b² and a % b = 63, find a and b. (SSC CGL 12 Sept, 2025 Shift - 3)
यदि a%b=a2−b2a \% b = a^2 - b^2a%b=a2−b2 तथा a%b=63a \% b = 63a%b=63 है, तो aaa और bbb के मान ज्ञात कीजिए। (SSC CGL 12 Sept, 2025 Shift - 3)
Shortcut Trick
Given:
If a % b = a² - b² and a % b = 63
Now, we check by options.
Option (c): 8, 1
8 - 1 = 64 - 1 = 63
So, the value of a = 8 and b = 1
Detailed Solution:
Given:
a%b=a2−b2a \% b = a^2 - b^2a%b=a2−b2
and
a%b=63a \% b = 63a%b=63
So,
a2−b2=63a^2 - b^2 = 63a2−b2=63
Now use identity:
a2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b)a2−b2=(a−b)(a+b)
⇒ (a−b)(a+b)=63(a-b)(a+b) = 63(a−b)(a+b)=63
Now take factors of 63:
63=7×963 = 7 \times 963=7×9
Let,
a−b=7a-b = 7a−b=7
a+b=9a+b = 9a+b=9
Add both equations:
2a=162a = 162a=16
⇒ a=8a = 8a=8
Substitute:
8−b=78 - b = 78−b=7
⇒ b=1b = 1b=1
∴ a = 8, b = 1
दिया है:
a%b=a2−b2a \% b = a^2 - b^2a%b=a2−b2
और
a%b=63a \% b = 63a%b=63
इसलिए,
a2−b2=63a^2 - b^2 = 63a2−b2=63
अब सूत्र लगाएँ:
a2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b)a2−b2=(a−b)(a+b)
⇒ (a−b)(a+b)=63(a-b)(a+b) = 63(a−b)(a+b)=63
63 के गुणनखंड लेते हैं:
63=7×963 = 7 \times 963=7×9
मान लेते हैं:
a−b=7a-b = 7a−b=7
a+b=9a+b = 9a+b=9
दोनों को जोड़ें:
2a=162a = 162a=16
⇒ a=8a = 8a=8
अब मान रखें:
8−b=78 - b = 78−b=7
⇒ b=1b = 1b=1
अंतिम उत्तर: a=8, b=1a = 8,\ b = 1a=8, b=1
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