CUET UG 2025 — General Test Quantitative Reasoning

Given: $7^{6n} - 6^{6n}$76n−66n, where $n$n is a positive integer

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Rewrite the expression:

$$7^{6n} - 6^{6n} = (7^6)^n - (6^6)^n$$76n−66n=(76)n−(66)n

Let $A = 7^6$A=76 and $B = 6^6$B=66

The expression becomes $A^n - B^n$An−Bn

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For any positive integer $n$n, the expression $A^n - B^n$An−Bn is always divisible by $(A - B)$(A−B).

Therefore, $(7^6)^n - (6^6)^n$(76)n−(66)n is always divisible by $(7^6 - 6^6)$(76−66).

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Calculate $7^6 - 6^6$76−66:

$$7^6 = 117,649$$76=117,649

$$6^6 = 46,656$$66=46,656

$$7^6 - 6^6 = 117,649 - 46,656 = 70,993$$76−66=117,649−46,656=70,993

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Testing divisibility of $70,993$70,993:

$70,993 \div 127 = 559$70,993÷127=559 (exact division)

$70,993 \div 556 = 127.69...$70,993÷556=127.69... (not exact)

$70,993 \div 17 = 4176.06...$70,993÷17=4176.06... (not exact)

$70,993 \div 23 = 3086.65...$70,993÷23=3086.65... (not exact)

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Since $(7^6)^n - (6^6)^n$(76)n−(66)n is always divisible by $(7^6 - 6^6) = 70,993$(76−66)=70,993, and $70,993$70,993 is divisible by $127$127, the expression $7^{6n} - 6^{6n}$76n−66n is always divisible by $127$127 for any positive integer $n$n.

Therefore, the answer is $127$127.