JOHN
Lv 7
JOHN asked in Science & MathematicsMathematics · 5 years ago

n is a 4-digit integer not divisible by 10, and Φ(n) is the integer whose digits are those of n in reverse order (eg, Φ(1234) = 4321).?

n is a 4-digit integer not divisible by 10, and Φ(n) is the integer whose digits are those of n in reverse order (eg, Φ(1234) = 4321). Find n such that Φ(n) = 4n + 3.

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  • Indica
    Lv 7
    5 years ago
    Favourite answer

    If number is abcd then dcba = 4(abcd) + 3 and all integers in [0,9] and d≥1

    RHS is odd so a must be odd

    a≤2 otherwise d>9 in LHS and so a=1. It follows that d≥4*1=4

    Comparing units on both sides gives a ≡ 4d+3 (mod10) → 2d+1 ≡ 0(mod5) → d=2,7

    d=7 is the only possibility and (i) gives 10²c+10b+7001 = 4(10²b+10c)+4*1007+3

    This reduces to 13b−2c = 99 so 11 | (b−c) and hence b=c=9

    Number is 1997

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